Compensation for geometric effects of beam misalignments in plane mirror interferometer metrology systems

ABSTRACT

In one aspect, the invention features a method, including using an interferometer in an interferometry system to produce an output beam comprising a phase related to an optical path difference between a path of a first beam and a path of a second beam, wherein the first beam contacts a measurement object at a first location and the first or second beam contacts the measurement object at a second location, and wherein the first and second locations are different, providing precalibrated information that accounts for contributions to the optical path difference caused by a deviation of the path of the first or second beam from a nominal beam path due to an imperfection of the measurement object at the first location and due to an imperfection of the measurement object at the second location, and determining a position of the measurement object with respect to at least one degree of freedom based on information derived from the output beam and the precalibrated information.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119 to ProvisionalPatent Application 60/479,741, entitled “COMPENSATION FOR GEOMETRICEFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETER METROLOGYSYSTEMS,” filed on Jun. 19, 2003, the entire contents of which is herebyincorporated by reference.

BACKGROUND

This invention relates to interferometers, e.g., linear and angulardisplacement measuring and dispersion interferometers, that measurelinear and angular displacements of a measurement object such as a maskstage or a wafer stage in a lithography scanner or stepper system, andalso interferometers that monitor wavelength and determine intrinsicproperties of gases.

Displacement measuring interferometers monitor changes in the positionof a measurement object relative to a reference object based on anoptical interference signal. The interferometer generates the opticalinterference signal by overlapping and interfering a measurement beamreflected from the measurement object with a reference beam reflectedfrom a reference object.

In many applications, the measurement and reference beams haveorthogonal polarizations and different frequencies. The differentfrequencies can be produced, for example, by laser Zeeman splitting, byacousto-optical modulation, or internal to the laser using birefringentelements or the like. The orthogonal polarizations allow a polarizingbeam-splitter to direct the measurement and reference beams to themeasurement and reference objects, respectively, and combine thereflected measurement and reference beams to form overlapping exitmeasurement and reference beams. The overlapping exit beams form anoutput beam that subsequently passes through a polarizer. The polarizermixes polarizations of the exit measurement and reference beams to forma mixed beam. Components of the exit measurement and reference beams inthe mixed beam interfere with one another so that the intensity of themixed beam varies with the relative phase of the exit measurement andreference beams.

A detector measures the time-dependent intensity of the mixed beam andgenerates an electrical interference signal proportional to thatintensity. Because the measurement and reference beams have differentfrequencies, the electrical interference signal includes a “heterodyne”signal having a beat frequency equal to the difference between thefrequencies of the exit measurement and reference beams. If the lengthsof the measurement and reference paths are changing relative to oneanother, e.g., by translating a stage that includes the measurementobject, the measured beat frequency includes a Doppler shift equal to 2νnp/λ, where ν is the relative speed of the measurement and referenceobjects, λ is the wavelength of the measurement and reference beams, nis the refractive index of the medium through which the light beamstravel, e.g., air or vacuum, and p is the number of passes to thereference and measurement objects. Changes in the phase of the measuredinterference signal correspond to changes in the relative position ofthe measurement object, e.g., a change in phase of 2π correspondssubstantially to a distance change L of λ/(2 np). Distance 2L is around-trip distance change or the change in distance to and from a stagethat includes the measurement object. In other words, the phase Φ,ideally, is directly proportional to L, and can be expressed asΦ=2pkL cos²θ  (1)for a plane mirror interferometer, e.g., a high stability plane mirrorinterferometer, where

$k = \frac{2\;\pi\; n}{\lambda}$and θ is the orientation of the measurement object with respect to anominal axis of the interferometer. This axis can be determined from theorientation of the measurement object where Φ is maximized. Where θ issmall, Equation (1) can be approximated byΦ=pkL(1−θ²)  (2)

Unfortunately, the observable interference phase, {tilde over (Φ)}, isnot always identically equal to phase Φ. Many interferometers include,for example, non-linearities such as those known as “cyclic errors.” Thecyclic errors can be expressed as contributions to the observable phaseand/or the intensity of the measured interference signal and have asinusoidal dependence on the change in for example optical path length 2pnL. A first order cyclic error in phase has, for example, a sinusoidaldependence on (4 πpnL)/λ and a second order cyclic error in phase has,for example, a sinusoidal dependence on 2(4 πpnL)/λ. Higher order cyclicerrors can also be present as well as sub-harmonic cyclic errors andcyclic errors that have a sinusoidal dependence of other phaseparameters of an interferometer system comprising detectors and signalprocessing electronics.

There are in addition to the cyclic errors, non-cyclic non-linearitiesor non-cyclic errors. An example of a source of a non-cyclic error isthe diffraction of optical beams in the measurement paths of aninterferometer. Non-cyclic error due to diffraction has been determinedfor example by analysis of the behavior of a system such as found in thework of J.-P. Monchalin, M. J. Kelly, J. E. Thomas, N. A. Kurnit, A.Szöke, F. Zernike, P. H. Lee, and A. Javan, “Accurate Laser WavelengthMeasurement With A Precision Two-Beam Scanning MichelsonInterferometer,” Applied Optics, 20(5), 736-757, 1981.

A second source of non-cyclic errors is the effect of “beam shearing” ofoptical beams across interferometer elements and the lateral shearing ofreference and measurement beams one with respect to the other. Beamshearing can be caused, for example, by a change in direction ofpropagation of the input beam to an interferometer or a change inorientation of the object mirror in a double pass plane mirrorinterferometer such as a differential plane mirror interferometer (DPMI)or a high stability plane mirror interferometer (HSPMI).

Accordingly, due to errors such as the aforementioned cyclic andnon-cyclic errors, the observable interference phase typically includescontributions in addition to Φ. Thus, the observable phase is moreaccurately expressed as{tilde over (Φ)}=Φ+ψ+ζ  (3)where ψ and ζ are the contributions due to the cyclic and non-cyclicerrors, respectively.

The effect of contributions to the observable phase due to cyclic andnon-cyclic errors can be reduced by quantifying these errors in eachinterferometer and correcting subsequent measurements with this data.Different techniques for quantifying cyclic errors are described incommonly owned U.S. Pat. No. 6,252,668, U.S. Pat. No. 6,246,481, U.S.Pat. No. 6,137,574, and U.S. patent application Ser. No. 10/287,898entitled “INTERFEROMETRIC CYCLIC ERROR COMPENSATION” filed Nov. 5, 2002by Henry A. Hill, the entire contents each of which are incorporatedherein by reference. In order to compensate for these contributions,cyclic error compensating systems and methods can be used to determine acyclic error function characterizing the cyclic error contribution tothe observed phase. Examples of apparatus and details of methods thatcan be used to characterize non-cyclic errors in interferometers andinterferometer components are described in U.S. patent application Ser.No. 10/366,587 entitled “CHARACTERIZATION AND COMPENSATION OF NON-CYCLICERRORS IN INTERFEROMETRY SYSTEMS,” to Henry A. Hill, filed on Feb. 12,2003, the entire contents of which are incorporated herein by reference.

Assuming any contributions due to cyclic and/or non-cyclic errors aresmall or otherwise compensated, according to Equation (2) the observablephase measured by a displacement measuring interferometer should beequal to 2 pkL(1−θ²). This relationship assumes that the optical pathdifference between the measurement and reference beam is equal to 2pkL(1−θ²) and allows one to readily determine L, a displacement of themeasurement object from the interferometer, from the measured phase,provided the orientation of the measurement object is known.

In the case of an interferometric metrology system including two or morelinear displacement plane mirror interferometers used, in part, tomeasure a change in angular orientation of a measurement object, theobservable phases measured by the two or more interferometers shouldeach be of the form 2 pkL(1−θ²). The resulting differences of phasesobtained either optically or electronically can be used to compute achange in angular orientation of a measurement object common to the twoor more plane mirror interferometers, provided that the measurement axesof the two or more interferometers are parallel.

SUMMARY

In certain aspects, the invention is based on the realization that adeviation of one or more of the interferometer beams from a nominal beampath can cause the optical path difference to vary from the optical pathdifference assumed for Equation (2). Errors arising from such deviationsare referred to as geometric non-cyclic errors. Where such errors arise,using the relationship in Equation (2) to determine L from the measuredphase can provide erroneous results, which can be detrimental inapplications demanding a high level of precision. Furthermore, effectsof such beam path deviations are heightened where L is comparativelylarge (e.g., 0.5 m or more) because the contribution of a beam pathdeviation to the optical path difference typically scales with L.

In some aspects, the invention is based on the realization that inmultiple pass interferometers (e.g., double pass interferometers), acompensation scheme that accounts for geometric errors associated withimperfections in the measurement object should account for imperfectionsat each location of the measurement object that is contacted by aninterferometer beam. For example, in system's using a double passinterferometer and a plane mirror measurement object, compensation formirror imperfections should account for imperfections at both mirrorlocations that are contacted by the interferometer measurement beam.

Compensation scheme's accounting for each point of contact of aninterferometer beam on the measurement object can account forimperfections of the measurement object which vary between the points ofcontact. In a double pass interferometer, for example, such acompensation scheme can account for measurement object imperfectionsthat cancel out each other's contribution to the measured phase. Incontrast, a compensation scheme which accounts only for mirrorimperfections at one location (e.g., along the interferometer axis), forexample, can provide erroneous compensation by failing to accommodatefor variations between the imperfections at each contact location of aninterferometer beam on the measurement object.

Imperfections in a measurement object can introduce a displacement ofthe measurement object from a nominal measurement object position.Alternatively, or additionally, imperfections in a measurement objectcan deflect an incident beam from a nominal beam path, which is the paththe beam would follow in the absence of any imperfections in themeasurement object. In some aspects, the invention is based on therealization that to accurately compensate for imperfections in themeasurement object, a compensation scheme should account for beamdeflections both within and out of the plane of incidence (i.e., theplane defined by the nominal beam path and the measurement objectsurface normal).

According, in certain aspects, the invention features methods andsystems that accurately compensate an interferometer measurement forgeometric errors due to mirror imperfections. Embodiments can alsoinclude compensating interferometer measurements for imperfections inone or more optics of the interferometer and/or for imperfections in thelight source used by the interferometer.

In some aspects, the invention is based on the realization that in amultiple degree of freedom interferometry system, a deviation of one ormore of the interferometer measurement axes from nominally parallelmeasurement axes can cause errors in computed changes in a degree ofchanges of a common measurement object. For example, where aninterferometry system monitors an angular orientation of a stage mirrorby monitoring the displacement of the stage mirror along two nominallyparallel axes, misalignment of the axes introduces an error into themeasured angular orientation. Moreover, because the separation of theinterferometer axes varies as a function of the mirror's displacementalong the axes, the error will vary systematically with the displacementof the stage mirror along the axes. Where such errors arise, using therelationship in Equation (2) to determine respective values of L fromthe measured phase can provide erroneous results, which can bedetrimental in applications demanding a high level of precision.

Accordingly, in certain aspects, the invention features methods andsystems for compensating interferometer measurements to account formisalignment of interferometer axes in a multiple degree of freedominterferometer system. Moreover, in some aspects, the invention featuresmethods and systems for compensating interferometer measurements toaccount for deviations of beam paths from the nominal beam path of asingle interferometer or from the nominal beam paths of a multipledegree of freedom interferometer system. Deviations of interferometerbeams paths from the nominal path can be determined while calibratingthe interferometer or interferometer system prior to its use and/orduring periods where the interferometry system is off-line. Thisprecalibrated information is provided with the interferometer orinterferometry system, and used by the system to correct measurements sothat they account for the beam path deviations.

Sources of beam path deviations include imperfections in one or more ofthe optical components making up the interferometer or interferometersystem, imperfections in the measurement object, and instabilities inthe interferometer light source that may cause the path of theinterferometer or interferometer system input beam to vary, andmisalignments of input beams to respective interferometers of a multipledegree of freedom interferometer system. Beam path deviations can becharacterized with respect to a nominal path corresponding to aperfectly aligned, defect free system. The nominal path for themeasurement beam and measurement beam component of the output beamdepends on the angular orientation of the measurement object. In otherwords, the nominal path is the path for which the optical pathdifference corresponds identically to 2 pkL(1−θ²), where L and θ aremeasured with respect to a fixed reference co-ordinate system. Also thenominal path for the reference beam and reference beam component of theoutput beam depends on the angular orientation of the reference objectthat may also be variable.

The precalibrated information may be stored as a representation (e.g., alookup table or a functional representation) in an electronic datastorage medium (e.g., a memory chip or a disk), which is provided to theinterferometer's end user. A control algorithm that runs theinterferometer in its end use application accesses the information fromthe data storage medium, and compensates the interferometer measurementaccordingly.

Compensation may be performed on-line in real time or off-line.Application of the methods may be used to isolate effects of othererrors, such as non-cyclic errors due to wavefront errors and beamshear.

Interferometers using techniques disclosed herein may be used inlithography tools and beam writing systems.

Various aspects and features of the invention are summarized below.

In general, in a first aspect, the invention features a method, thatincludes using an interferometer in an interferometry system to producean output beam comprising a phase related to an optical path differencebetween a path of a first beam and a path of a second beam, wherein thefirst beam contacts a measurement object at a first location and thefirst or second beam contacts the measurement object at a secondlocation, and wherein the first and second locations are different,providing precalibrated information that accounts for contributions tothe optical path difference caused by a deviation of the path of thefirst or second beam from a nominal beam path due to an imperfection ofthe measurement object at the first location and due to an imperfectionof the measurement object at the second location, and determining aposition of the measurement object with respect to at least one degreeof freedom based on information derived from the output beam and theprecalibrated information.

Embodiments of the method may include one or more of the followingfeatures.

The precalibrated information can account for contributions to theoptical path difference caused by a deviation of the path of the firstor second beam within a plane defined by the nominal beam path due tothe imperfection of the measurement object at the first or secondlocation. The precalibrated information can account for contributions tothe optical path difference caused by a deviation of the path of thefirst or second beam out of a plane defined by a nominal beam path dueto the imperfection of the measurement object at the first or secondlocation. The precalibrated information can further account forcontributions to the optical path difference caused by a deviation ofthe path of the first or second beam from the nominal beam path due toan imperfection in at least one optic of the interferometer. Forexample, the imperfection in at least one optic of the interferometercan include an imperfection in a surface of the optic and/or a bulkimperfection in the optic. The precalibrated information can furtheraccount for contributions to the optical path difference caused by adeviation of the path of the first or second beam from the nominal beampath due to an imperfection in a light source that causes an input beamderived from the light source to deviate from an input beam path to theinterferometer.

The first or second beam can contact the measurement object at one ormore additional locations different from the first and second locationsand the precalibrated information can account for contributions to theoptical path difference caused by a deviation of the path of the firstor second beam from the nominal beam path due to an imperfection of themeasurement object at one or more additional locations.

The precalibrated information can be parameterized in terms of at leastone of an angular orientation of the measurement object relative to theinterferometer, a distance between the measurement object and theinterferometer, and a direction of an input beam to the interferometer.In some embodiments, the precalibrated information is parameterized interms of at least two of an angular orientation of the measurementobject relative to the interferometer, a distance between themeasurement object and the interferometer, and a direction of an inputbeam to the interferometer. For example, the precalibrated informationcan be parameterized in terms of an angular orientation of themeasurement object relative to the interferometer, a distance betweenthe measurement object and the interferometer, and a direction of aninput beam to the interferometer. The precalibrated information can bestored as a representation in an electronic storage medium. For example,the representation can include a lookup table and/or a functionalrepresentation.

The determined position of the measurement object with respect to atleast one degree of freedom can be related to a displacement of themeasurement object relative to the interferometer. For example, thedetermined position of the measurement object with respect to at leastone degree of freedom can be the displacement of the measurement objectrelative to the interferometer. The determined position of themeasurement object with respect to at least one degree of freedom can berelated to an angular orientation of the measurement object. Forexample, the determined position of the measurement object with respectto at least one degree of freedom can be the angular orientation of themeasurement object.

Determining the position of the measurement object can include measuringthe phase of the output beam and relating the phase to the position ofthe measurement object based on one or more values derived from thepredetermined information. The values derived from the predeterminedinformation can be selected based on the phase. Te values derived fromthe predetermined information can be selected based on an angularorientation of the measurement object. The values derived from thepredetermined information can be selected based on a path of an inputbeam derived from the light source to the interferometer relative to thenominal path. The method can further include monitoring deviations ofthe input beam path from the nominal path. A relationship between thephase, Φ, and the optical path difference can be expressed by theequationΦ=2pkLζ(1−(θ₁−η₁)²−(θ₂−η₂)²)+2k(X ₁ +X ₂),wherein p is an integer, k is a wavenumber, L is a relative distancebetween the interferometer and the measurement object, θ₁ and θ₂ areangular orientation of the measurement object with respect to theinterferometer along orthogonal coordinates, and ζ and η₁ and η₂ areterms that depend on the deviation of at least one of the beam pathsfrom the nominal beam path, and X₁ and X₂ is are local displacements ofa surface of the measurement object from a nominal plane surface at thefirst and second locations, respectively.

Using the interferometer to produce the output beam can includeproducing the output beam as the measurement object is moved relative tothe interferometer, and determining the position of the measurementobject can include monitoring the position of the measurement objectduring the relative movement. Using the interferometer to produce theoutput beam can include separating an input beam into at least the firstand second beams, directing the first and second beams along theirrespective paths, and recombining the two beams after one or both of thebeams contacts the measurement object. The second beam can contact themeasurement object at the second location and the optical pathdifference can be related to an angular orientation of the measurementobject with respect to the interferometer.

In general, in another aspect, the invention features a method, thatincludes using an interferometer to produce an output beam comprising aphase related to an optical path difference between a path of a firstbeam and a path of a second beam, wherein the first beam contacts ameasurement object at a first location, providing precalibratedinformation that accounts for contributions to the optical pathdifference caused by a deviation of the path of the first beam out of aplane defined by a nominal beam path due to an imperfection of themeasurement object at the first location, and determining a position ofthe measurement object with respect to at least one degree of freedombased on information derived from the output beam and the precalibratedinformation.

Embodiments of the method may include one or more of the followingfeatures and/or features of other aspects.

The precalibrated information can account for contributions to theoptical path difference caused by a deviation of a path of the firstbeam out within the plane defined by the nominal beam path due to theimperfection of the measurement object at the first location. Theprecalibrated information can account for contributions to the opticalpath difference caused by a local displacement of a surface of themeasurement object from a nominal plane surface at the first location.The first or second beam can contact the measurement object at a secondlocation, wherein the first and second locations are different. Theprecalibrated information can account for contributions to the opticalpath difference caused by a deviation of the path of the first or secondbeam from a nominal beam path due to an imperfection of the measurementobject at the second location.

In general, in a further aspect, the invention features a method, thatincludes using an interferometer to produce an output beam comprising aphase related to an optical path difference between a first beam pathand a second beam path, wherein the first or second beam contacts ameasurement object, providing precalibrated information that accountsfor contributions to the optical path difference caused by a deviationof a path of the first beam from a nominal beam path due to animperfection of the measurement object, and accounts for contributionsto the optical path difference caused by a deviation of the path of thefirst beam from the nominal beam path due to an imperfection in one ormore optics of the interferometer or in a light source used to producethe output beam, and determining a position of the measurement objectwith respect to at least one degree of freedom based on informationderived from the output beam and the precalibrated information.

Embodiments of the method may include one or more of the followingfeatures and/or features of other aspects.

In general, in another aspect, the invention features a method, thatincludes using a first interferometer and a second interferometer in aninterferometry system to produce a first output beam and a second outputbeam, respectively, wherein each output beam comprises a phase relatedto an optical path difference between two beam paths, at least one ofwhich contacts a measurement object, providing precalibrated informationthat accounts for a misalignment of an axis of the first interferometerrelative to an axis of the second interferometer, and determining aposition of the measurement object with respect to at least one degreeof freedom based on information derived from the first and second outputbeams and the precalibrated information.

Embodiments of the method may include one or more of the followingfeatures and/or features of other aspects.

The degree of freedom can correspond to an angular orientation of themeasurement object. For each interferometer, the precalibratedinformation can account for contributions to the optical path differencecaused by a deviation of at least one of the beam paths from a nominalbeam path due to other imperfections in the interferometry system. Otherimperfections can include an imperfection in at least one optic of theinterferometer, an imperfection in the measurement object, or animperfection in a light source that causes an input beam derived fromthe light source to deviate from an input beam path for theinterferometer.

In general, in another aspect, the invention features an apparatusincluding an interferometer configured to produce an output beamcomprising a phase related to an optical path difference between a pathof a first beam and a path of a second beam, wherein the first beamcontacts a measurement object at a first location and the first orsecond beam contacts the interferometer at a second location, andwherein the first and second locations are different, and an electroniccontroller coupled to the interferometer, wherein during operation theelectronic controller determines a position of the measurement objectwith respect to at least one degree of freedom based on informationderived from the output beam and precalibrated information that accountsfor contributions to the optical path difference caused by a deviationof at least one of the beam paths from a nominal beam path due to animperfection of the measurement object at the first location and due toan imperfection of the measurement object at the second location.

Embodiments of the apparatus may include one or more of the features ofother aspects.

In general, in another aspect, the invention features an apparatus,including an interferometer configured to produce an output beamcomprising a phase related to an optical path difference between a pathof a first beam and a path of a second beam, wherein the first beamcontacts a measurement object at a first location, and an electroniccontroller coupled to the interferometer, wherein during operation theelectronic controller determines a position of the measurement objectwith respect to at least one degree of freedom based on informationderived from the output beam and precalibrated information that accountsfor contributions to the optical path difference caused by a deviationof the path of the first beam out of a plane defined by a nominal beampath due to an imperfection of the measurement object at the firstlocation.

Embodiments of the apparatus may include one or more of the features ofother aspects.

In general, in another aspect, the invention features an interferometrysystem, including a measurement object, a first interferometer and asecond interferometer, the first and second interferometers respectivelybeing configured to produce an output beam comprising a phase related toan optical path difference between a path of a first beam and a path ofa second beam, wherein the first beam contacts the measurement object ata first location; and an electronic controller coupled to theinterferometer, wherein during operation the electronic controllerdetermines a position of the measurement object with respect to at leastone degree of freedom based on information derived from the output beamand precalibrated information that accounts for a misalignment of anaxis of the first interferometer relative to an axis of the secondinterferometer.

Embodiments of the system may include one or more of the features ofother aspects.

In another aspect, the invention features a lithography system for usein fabricating integrated circuits on a wafer, the system including astage for supporting the wafer, an illumination system for imagingspatially patterned radiation onto the wafer, a positioning system foradjusting the position of the stage relative to the imaged radiation,and a foregoing apparatus for monitoring the position of the waferrelative to the imaged radiation.

In another aspect, the invention features a lithography system for usein fabricating integrated circuits on a wafer, the system including astage for supporting the wafer, and an illumination system including aradiation source, a mask, a positioning system, a lens assembly, and aforegoing apparatus, wherein during operation the source directsradiation through the mask to produce spatially patterned radiation, thepositioning system adjusts the position of the mask relative to theradiation from the source, the lens assembly images the spatiallypatterned radiation onto the wafer, and the apparatus monitors theposition of the mask relative to the radiation from the source.

In a further aspect, the invention features a beam writing system foruse in fabricating a lithography mask, the system including a sourceproviding a write beam to pattern a substrate, a stage supporting thesubstrate, a beam directing assembly for delivering the write beam tothe substrate, a positioning system for positioning the stage and beamdirecting assembly relative one another, and a foregoing apparatus formonitoring the position of the stage relative to the beam directingassembly.

In another aspect, the invention features a lithography method for usein fabricating integrated circuits on a wafer, the method includingsupporting the wafer on a moveable stage, imaging spatially patternedradiation onto the wafer, adjusting the position of the stage, andmonitoring the position of the stage using the a foregoing method.

In a further aspect, the invention features a lithography method for usein the fabrication of integrated circuits including directing inputradiation through a mask to produce spatially patterned radiation,positioning the mask relative to the input radiation, monitoring theposition of the mask relative to the input radiation using a foregoingmethod, and imaging the spatially patterned radiation onto a wafer.

In yet another aspect, the invention features a lithography method forfabricating integrated circuits on a wafer including positioning a firstcomponent of a lithography system relative to a second component of alithography system to expose the wafer to spatially patterned radiation,and monitoring the position of the first component relative to thesecond component using a foregoing method.

In a further aspect, the invention features a method for fabricatingintegrated circuits, the method including a foregoing lithographymethod.

In another aspect, the invention features a method for fabricatingintegrated circuits, the method including using a foregoing lithographysystem.

In yet another aspect, the invention features a method for fabricating alithography mask, the method including directing a write beam to asubstrate to pattern the substrate, positioning the substrate relativeto the write beam, and monitoring the position of the substrate relativeto the write beam using the a foregoing interferometry method.

Embodiments of the invention may include one or more of the followingadvantages.

Characterizing beam path deviations in interferometers and/orinterferometer components using the techniques disclosed herein can beused to improve interferometer accuracy in end-use applications.Accuracy improvement comes from compensating for the contribution ofbeam path deviations to the optical path difference when determining adegree of freedom (e.g., a displacement or angular orientation) of ameasurement object from the interference phase. This also can allow forthe use of interferometer and interferometer components in highprecision applications where imperfections causing beam path deviationswould otherwise render the interferometer and/or components tooinaccurate. Accordingly, interferometers and/or components can be usedin applications that would otherwise require higher quality componentsto provide a desired level of accuracy. Because lesser qualitycomponents are typically cheaper than high quality counterparts, thetechniques can provide a cost savings.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. In case of conflict withpublications, patent applications, patents, and other referencesmentioned incorporated herein by reference, the present specification,including definitions, will control.

Other features, objects, and advantages of the invention will beapparent from the following detailed description.

DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic diagram of an embodiment of an interferometrysystem.

FIG. 2 is a schematic diagram of a high stability plane mirrorinterferometer HSPMI.

FIG. 3( a) is a schematic diagram showing beam path deviations for beamsin the HSPMI of FIG. 2.

FIG. 3( b) is a diagram showing local surface properties of themeasurement object in the HSPMI of FIG. 2.

FIG. 4 is a schematic diagram of an embodiment of an angulardisplacement interferometer.

FIG. 5 is a schematic diagram showing the path of a beam throughportions of the angular displacement interferometer shown in FIG. 4.

FIG. 6 and FIG. 7 are schematic diagrams showing the path of a beamthrough other portions of the angular displacement interferometer shownin FIG. 4.

FIG. 8 is a schematic diagram of an embodiment of a beam shearingassembly.

FIG. 9 is a schematic diagram of an embodiment of a multiple degree offreedom interferometer.

FIG. 10 is a schematic diagram of an embodiment of a lithography toolthat includes an interferometer.

FIG. 11( a) and FIG. 11( b) are flow charts that describe steps formaking integrated circuits.

FIG. 12 is a schematic of a beam writing system that includes aninterferometry system.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

Referring to FIG. 1, an interferometry system 100 includesinterferometer subsystems 101 and 151 that are respectively configuredto monitor a displacement of a plane mirror measurement object 190 and abeam propagation direction. Subsystem 101 includes an interferometer 110positioned to receive an input beam 122 from a source 115.Interferometer 110 splits input beam 122 into a measurement beam 121 anda reference beam (not shown), directs measurement beam 121 and thereference beam along different paths, and recombines them to form anoutput beam 123. Interferometer 110 directs measurement beam 121 toreflect from measurement object 190. Although measurement beam 121 isdepicted as making a single pass between measurement object 190 andinterferometer 110, in many embodiments it makes multiple passes to themeasurement object. Output beam 123 impinges on a detector 120, whichdetects intensity variations in a polarization component of output beam123. Detector 120 communicates the time-varying intensity variations toan electronic controller 140 as an interference signal from whichelectronic controller 140 extracts an interference phase. Theinterference phase is related to the optical path difference betweenmeasurement beam 121 and the reference beam. Subsequently, electronicprocessor 140 determines a displacement of measurement object 190relative to interferometer 110 based on a known relationship between thephase, the optical path difference and the relative displacement.

Subsystem 151 includes an angle interferometer 150 and a detector 155. Abeam-splitter 145 directs a portion 152 of the beam from light source115 towards angle interferometer 150, which generates an output beam 153having an interference phase related to the direction of input beam 122.Detector 155 monitors the intensity of a polarization component ofoutput beam 153 and communicates an interference signal related to theoutput beam intensity to electronic controller 140. Electroniccontroller 140 then extracts an interference phase from the interferencesignal and determines deviations of the propagation direction of inputbeam 122 from variations of the interference phase.

In FIGS. 1-3( b), reference is made to a Cartesian co-ordinate systemsuch as shown in FIG. 1.

Optimally, the paths of input beam 122, measurement beam 121, and thecomponents of output beam 123 coincide with a nominal beam path. Thenominal path corresponds to the path of the beam where the input beamhas a fixed orientation relative to a preferred measurement axis ofinterferometer 110 and the optics making up interferometer 110 and planemirror measurement object 190 are perfect. The nominal path of themeasurement beam (and the measurement beam component of the output beam)is determined according to the orientation of mirror 190 with respect tothe interferometer measurement axis. The interferometer measurement axisis parallel to the x-axis. Accordingly, rather than there being onenominal path for the measurement beam, there is a different nominal pathfor each orientation of the measurement object at each displacement ofthe measurement object relative to the interferometer.

Due to imperfections in one or more of the optical components (e.g.,deviations in the flatness of an optical surface or refractive indexvariations of a component) or instabilities or misalignment of source115, the path of input beam 122, measurement beam 121, and/or thecomponents of output beam 123 may deviate from the nominal beam path.These deviations can cause the optical path difference between themeasurement beam and reference beam to vary from the optical pathdifference implied by a relationship such as given by Equation (2) abovefor a plane mirror interferometer. System 100 accounts for contributionsto the optical path difference caused by these deviations by accessing alookup table that provides correction data parameterized as a functionof one or more measurable system parameters, and determines thedisplacement of the measurement object relative to interferometer 110based on a corrected optical path difference. Determination of thecorrect optical path difference is described in detail for a specificinterferometer below.

As an example, in some embodiments, interferometer 110 is a highstability plane mirror interferometer (HSPMI). Referring to FIG. 2, anHSPMI 111 includes a polarization beam-splitter 30, a retroreflector 32,quarter wave phase retardation plates 34 and 36, and a plane mirrorreference object 42. Input beam 122 is a two-component beam. The twocomponents have different frequencies and are orthogonally planepolarized. The different frequencies can be produced in source 115, forexample, by laser Zeeman splitting, by acousto-optical modulation, orinternal to the laser using birefringent elements or the like. HSPMI 111splits input beam 122 into two components. One component, shown as firstand second pass measurement beams 22 and 24, reflects from the surfaceof measurement object 190 twice before exiting HSPMI 111. The othercomponent, shown by first and second pass reference beams 28 and 27,reflect from reference mirror 42 twice before exiting HSPMI 111. Theexiting beam components overlap and form output beam 123.

An electrical interference signal 52 is generated by the detection ofoutput beam 123 in detector 120. Detector 120 includes a polarizer tomix the reference and measurement beam components of output beam 123with respect to polarization. Electrical interference signal 52 containsa heterodyne signal having a heterodyne phase Φ.

Reference is made to FIG. 3( a) in discussing the relationship between aphysical displacement L to be measured by HSMPI 111 and the measuredphase Φ of the heterodyne signal from HSPMI 111. Phase Φ is expressed asthe sum of three phases Φ_(M), Φ_(S), and Φ_(R), i.e.,Φ=Φ_(M)+Φ_(S)+Φ_(R)  (4)where Φ_(M) is the contribution to phase Φ from the measurement beampath in HSMPI 111, Φ_(S) is the phase contribution introduced by arelative lateral shear of the measurement and reference beam componentsof output beam 123 and a relative difference in directions ofpropagation of the measurement and reference beam components of outputbeam 123, and Φ_(R) is the contribution to phase Φ from the referencebeam path in HSMPI 111. With reference to FIGS. 3( a) and 3(b), phasesΦ_(M) and Φ_(S) are given to a good approximation by the equations

$\begin{matrix}{{\Phi_{M} = {{{kL}\mspace{11mu}\cos\mspace{11mu}\theta_{z}\mspace{11mu}\cos\mspace{11mu}\theta_{y} \times \begin{Bmatrix}{\frac{\left\lbrack {1 + {\cos\; 2\;\left( {\theta_{z} + \theta_{z1} + \alpha_{z}} \right)\mspace{11mu}\cos\; 2\;\left( {\theta_{y} + \theta_{y1} + \alpha_{y}} \right)}} \right\rbrack}{\cos\;\left( {\theta_{z} + {2\;\theta_{z1}} + \alpha_{z}} \right)\mspace{11mu}{\cos\left( {\theta_{y} + {2\;\theta_{y1}} + \alpha_{y}} \right)}} +} \\\frac{\begin{Bmatrix}{1 + {\cos\;{2\left\lbrack {\theta_{z} + \theta_{z1} + \left( {\theta_{z1} - \theta_{z2}} \right) + \alpha_{z} + \beta_{z}} \right\rbrack}} +} \\{\cos\;{2\left\lbrack {\theta_{y} + \theta_{y1} + \left( {\theta_{y1} - \theta_{y2}} \right) + \alpha_{y} + \beta_{y}} \right\rbrack}}\end{Bmatrix}}{\cos\;\left( {\theta_{z} + {2\;\theta_{z1}} + \alpha_{z} + \beta_{z}} \right)\mspace{11mu}\cos\;\left( {\theta_{y} + {2\;\theta_{y1}} + \alpha_{y} + \beta_{y}} \right)}\end{Bmatrix}} + {2\left( {X_{1} + X_{2}} \right)}}},} & (5) \\{\Phi_{S} = {\frac{\xi\; L}{2}\begin{Bmatrix}{\left\lbrack {{2\left( {\theta_{z1} - \theta_{z2}} \right)} - {2\left( {\theta_{z1} - \theta_{z2}} \right)_{R}} + \left( {\alpha_{z} - \alpha_{zR}} \right) + \left( {\gamma_{z} - \gamma_{zR}} \right)} \right\rbrack \times} \\{\left\lbrack {{4\;\theta_{z}} + {4\;\theta_{z1}} + {2\left( {\theta_{z1} - \theta_{z2}} \right)} + {4\;\alpha_{z}} + {2\;\beta_{z}}} \right\rbrack +} \\{\left\lbrack {{2\left( {\theta_{y1} - \theta_{y2}} \right)} - {2\left( {\theta_{x1} - \theta_{x2}} \right)_{R}} + \left( {\alpha_{y} - \alpha_{yR}} \right) + \left( {\gamma_{y} - \gamma_{yR}} \right)} \right\rbrack \times} \\\left\lbrack {{4\;\theta_{y}} + {4\;\theta_{y1}} + {2\left( {\theta_{y1} - \theta_{y2}} \right)} + {4\;\alpha_{y}} + {2\;\beta_{y}}} \right\rbrack\end{Bmatrix}}} & (6)\end{matrix}$where L is the distance between point, N (which depends on therefractive indices of the media in the measurement beam path), theconjugate of the nodal point of retroreflector 32 as seen throughpolarization beam-splitter 30, and surface 190A of object mirror 190along a measurement axis as defined for interferometer 111; L_(R) is thedistance between point N and the reference mirror 42 along a measurementaxis as defined for interferometer 111; k is a wavenumber correspondingto wavelength λ of source 115; θ_(z) and θ_(y) are the rotations ofsurface 190A of object mirror 190 about z and y axes, respectively;θ_(z1) and θ_(z2) are the local slopes of surface 190F of object mirror190 measured in the x-y plane at the positions where the first andsecond pass measurement beams 22 and 24, respectively, contact objectmirror 190; θ_(z1) and θ_(z2) are the local slopes of surface 190F ofobject mirror 190 measured in the x-z plane at the positions where thefirst and second pass measurement beams 22 and 24, respectively, contactobject mirror 190; α_(z) and α_(y) are deviations in the direction ofthe input beam with respect to the interferometer measurement axis inthe x-y and x-z planes, respectively; β_(z) and β_(y) are deviations inthe direction of the component of second pass measurement beam 24propagating toward object mirror 190 with respect to the direction ofpropagation of the component of the first pass measurement beam 22propagating toward HSMPI 111 in the x-y and x-z planes, respectively;γ_(z) and γ_(y) are deviations in the direction of the measurement beamcomponent of output beam 123 with respect to the direction of componentof the second pass measurement beam 24 propagating toward HSPMI 111 inthe x-y and x-z planes, respectively; the terms X₁ and X₂ are the localdisplacements surface 190F from surface 190A of object mirror 190measured in the x-y plane at the positions where the first and secondpass measurement beams 22 and 24, respectively, contact object mirror190; and ξ is one plus the ratio of the measurement beam path lengthbetween HSPMI 111 and the photosensitive surface of detector 120 and themeasurement beam path length between HSMPI 111 and measurement object190 (which is L). The terms that have a subscript R represent parametersassociated with the reference beam, e:g., the term (θ_(z))_(R) inEquation (6) represents the rotation about the z axis of the conjugateof reference mirror 42 located in the space of measurement beam 121 andnominally parallel to surface 190A of object mirror 190.

The location of the conjugate of the nodal point of retroreflector 32 isdisplaced from the conjugate of the apex of retroreflector 32 as seenthrough polarization beam-splitter 30 depending on the physical pathlength of the measurement path in polarization beam-splitter 30 andretroreflector 32 and magnitude of the index of refraction ofpolarization beam-splitter 30 and retroreflector 32. The nodal pointrefers to the location of the image of the vertex of retroreflector 32as seen from outside the retroreflector. The value for L may be writtenfor example as

$\begin{matrix}{L = {{n_{a}\mspace{11mu} l_{M}} + \frac{n_{a}^{2}\mspace{11mu} l_{M,I}}{n_{I}}}} & (7)\end{matrix}$where l_(M) is the one way physical path length of the measurement pathin air for θ_(y)=0 and θ_(z)=0, n_(I) is the refractive index of theglass portion of an interferometer comprising a single medium, andl_(M,I) is the one way physical path length of the measurement beam inthe glass portion of the interferometer for θ_(y)=0 and θ_(z)=0. Thelength l_(M,I) corresponds to the respective physical path length of abeam at θ_(y)=0 and θ_(z)=0, respectively, measured from the apex ofretroreflector 32.

Surface 190F of object mirror 190 corresponds to the physical surface ofobject mirror 190 and can be characterized by techniques describedsubsequently. Surface 190A of object mirror 190 represents an average ofthe physical surface 190F according to an algorithm such as aleast-squares fit. In preferred embodiments, the characterization of theobject mirror surface has a resolution on the order of, or greater than,the resolution of the measurement beam diameter.

An equation for Φ_(R) is of the same general form as that of theequation given for Φ_(M), i.e., Equation (5). The corresponding equationΦ_(R) may be used to evaluate non-linear errors that arise from thereference beam path in an end use applications such as an interferometerconfigured with a column reference, where a polarization leakage filteris used such as described in Provisional Patent Application No.60/303,299 entitled “INTERFEROMETRY SYSTEM AND METHOD EMPLOYING ANANGULAR DIFFERENCE IN PROPAGATION BETWEEN ORTHOGONALLY POLARIZED INPUTBEAM COMPONENTS,” to Peter de Groot et al. and its corresponding utilityapplication U.S. patent application Ser. No. 10/174,149, or whereelements of the interferometer are rotated or tilted to eliminatecertain cyclic non-linear errors such as described in Provisional PatentApplication No. 60/314,490 entitled “TILTED INTERFEROMETER” to Henry A.Hill and its corresponding utility application U.S. patent applicationSer. No. 10/218,965. The contents of both cited Provisional PatentApplications and both Utility U.S. patent applications are herebyincorporated by reference in their entirety.

The differences δ_(z) and δ_(y) in the directions of propagation ofmeasurement and reference beam components of output beam 123 in the x-yand x-z planes, respectively, areδ_(z)=[2(θ_(z1)−θ_(z2))−2(θ_(z1)−θ_(z2))_(R)+(α_(z)−α_(zR))+(γ_(z)−γ_(zR))],  (8)δ_(y)=[2(θ_(y1)−θ_(y2))−2(θ_(x1)−θ_(x2))_(R)+(α_(y)−α_(yR))+(γ_(y)−γ_(yR))].  (9)Note, for example, that where θ_(z1)=θ_(z2), the contributions to δ_(z)due to variations in the slope of the stage mirror in the x-y plane atthe points at which the measurement beam contacts the stage cancel eachother out.

The subsequent description of Equation (4) is in terms of thecontributions that arise from Φ_(R) and Φ_(S) since, as noted above, thedescription of the contribution of Φ_(R) is the same as thecorresponding portion of the description of the contribution of Φ_(M).The contributions of Φ_(R) and Φ_(S) given by Equations (5) and (6) maybe expanded in a power series as

$\begin{matrix}{\frac{\Phi_{M} + \Phi_{S}}{4k} = {{+ {L\begin{bmatrix}{1 - \left( {\theta_{z}^{2} + \theta_{y}^{2}} \right) - {\theta_{z}\left\lbrack {{- {\xi\left( {\theta_{z1} - \theta_{z2}} \right)}} + \alpha_{z} + \frac{\beta_{z}}{2} - {\xi\;\frac{\delta_{z}}{2}}} \right\rbrack} -} \\{{\theta_{y}\left( {{- {\xi\left( {\theta_{y1} - \theta_{y2}} \right)}} + \alpha_{y} + \frac{\beta_{y}}{2} - \frac{\delta_{y}}{2}} \right)} +} \\{\theta_{z1}^{2} - {\left( {\theta_{z1} - \theta_{z2}} \right)\left( {\theta_{z1} + \alpha_{z} + \beta_{z}} \right)} - \left( {\theta_{z1} - \theta_{z2}} \right)^{2} +} \\{\theta_{y1}^{2} - {\left( {\theta_{y1} - \theta_{y2}} \right)\left( {\theta_{y1} + \alpha_{y} + \beta_{y}} \right)} - \left( {\theta_{y1} - \theta_{y2}} \right)^{2} +} \\{{\frac{1}{4}{{\xi\left\lbrack {{2\left( {\theta_{z1} - \theta_{z2}} \right)} + \delta_{z}} \right\rbrack}\left\lbrack {{2\;\theta_{z1}} + \left( {\theta_{z1} - \theta_{z2}} \right) + {2\;\alpha_{z}} + \beta_{z}} \right\rbrack}} +} \\{{\frac{1}{4}{{\xi\left\lbrack {{2\left( {\theta_{y1} - \theta_{y2}} \right)} + \delta_{y}} \right\rbrack}\left\lbrack {{2\;\theta_{y1}} + \left( {\theta_{y1} - \theta_{y2}} \right) + {2\alpha_{y}} + \beta_{y}} \right\rbrack}} -} \\{{\frac{1}{4}\left( {\alpha_{z}^{2} + \alpha_{y}^{2}} \right)} - {\frac{1}{4}\left( {\alpha_{z} + \beta_{z}} \right)^{2}} - {\frac{1}{4}\left( {\alpha_{y} + \beta_{y}} \right)^{2}} + \ldots}\end{bmatrix}}} + {\frac{1}{2}\left( {X_{1} + X_{2}} \right)}}} & (10)\end{matrix}$wherein the leading terms have been retained up through quadratic terms.In order to make the contributions of the deviations from the nominalpath more easily identifiable, Equation (10) may be rewritten as

$\begin{matrix}{\frac{\Phi_{M} + \Phi_{S}}{k} = {{{+ 4}L\begin{Bmatrix}{1 - \left\{ {\theta_{z} + {\frac{1}{2}\left\lbrack {{- \left( {\theta_{z1} - \theta_{z2}} \right)} + \alpha_{z} + \frac{\beta_{z}}{2} - {\xi\;\frac{\delta_{z}}{2}}} \right\rbrack}} \right\}^{2} -} \\{\left\{ {\theta_{y} + {\frac{1}{2}\left\lbrack {{- \left( {\theta_{y1} - \theta_{y2}} \right)} + \alpha_{y} + \frac{\beta_{y}}{2} - {\xi\;\frac{\delta_{y}}{2}}} \right\rbrack}} \right\}^{2} +} \\{\theta_{z1}^{2} - {\left( {\theta_{z1} - \theta_{z2}} \right)\left\lbrack {{\left( {1 - \xi} \right)\theta_{z1}} + {\left( {1 - \frac{\xi}{2}} \right)\alpha_{z}} +} \right.}} \\{\left. {{\left( {1 - \frac{\xi}{4}} \right)\beta_{z}} - {\frac{\xi}{4}\left( {1 + \xi} \right)\delta_{z}}} \right\rbrack +} \\{\theta_{y1}^{2} - {\left( {\theta_{y1} - \theta_{y2}} \right)\left\lbrack {{\left( {1 - \xi} \right)\theta_{y1}} + {\left( {1 - \frac{\xi}{2}} \right)\alpha_{y}} +} \right.}} \\{\left. {{\left( {1 - \frac{\xi}{4}} \right)\beta_{y}} - {\frac{\xi}{4}\left( {1 + \xi} \right)\delta_{y}}} \right\rbrack -} \\{{\left( {\theta_{z1} - \theta_{z2}} \right)^{2}\left( {\frac{1}{2} - \frac{\xi}{2} - \frac{\xi^{2}}{4}} \right)} - {\left( {\theta_{y1} - \theta_{y2}} \right)^{2}\left( {\frac{1}{2} - \frac{\xi}{2} - \frac{\xi^{2}}{4}} \right)} -} \\{{\frac{1}{4}\left( {\alpha_{z} + \frac{\beta_{z}}{2} - {\xi\;\frac{\delta_{z}}{2}}} \right)^{2}} - {\frac{1}{4}\left( {\alpha_{y} + \frac{\beta_{y}}{2} - {\xi\;\frac{\delta_{y}}{2}}} \right)^{2}} -} \\{{\frac{1}{8}\left( {\beta_{z}^{2} + \beta_{y}^{2}} \right)} + {\frac{\xi^{2}}{8}\left( {\delta_{z}^{2} + \delta_{y}^{2}} \right)} + \ldots}\end{Bmatrix}} + {2{\left( {X_{1} + X_{2}} \right).}}}} & (11)\end{matrix}$

According to Equation (11), the effects of the deviations are equivalentto a change in the directions of the effective measurement axis in thex-y and x-z planes by η_(z) and η_(y), respectively, with

$\begin{matrix}{{\eta_{z} = {\frac{1}{2}\left\lbrack {{- \left( {\theta_{z1} - \theta_{z2}} \right)} + \alpha_{z} + \frac{\beta_{z}}{2} - {\xi\;\frac{\delta_{z}}{2}}} \right\rbrack}},} & (12) \\{\eta_{y} = {\frac{1}{2}\left\lbrack {{- \left( {\theta_{y1} - \theta_{y2}} \right)} + \alpha_{y} + \frac{\beta_{y}}{2} - {\xi\;\frac{\delta_{y}}{2}}} \right\rbrack}} & (13)\end{matrix}$and to change the effective scale or equivalent wavelength by a factor ζwhere

$\begin{matrix}\begin{matrix}{\zeta = {1 -}} \\{\begin{bmatrix}{\theta_{z1}^{2} - {\left( {\theta_{z1} - \theta_{z2}} \right)\left\lbrack {{\left( {1 - \xi} \right)\theta_{z1}} + {\left( {1 - \frac{\xi}{2}} \right)\alpha_{z}} +} \right.}} \\{\left. {{\left( {1 - \frac{\xi}{4}} \right)\beta_{z}} - {\frac{\xi}{4}\left( {1 - \xi} \right)\delta_{z}}} \right\rbrack +} \\{\theta_{y1}^{2} - {\left( {\theta_{y1} - \theta_{y2}} \right)\left\lbrack {{\left( {1 - \xi} \right)\theta_{y1}} + {\left( {1 - \frac{\xi}{2}} \right)\alpha_{y}} +} \right.}} \\{\left. {{\left( {1 - \frac{\xi}{4}} \right)\beta_{y}} - {\frac{\xi}{4}\left( {1 + \xi} \right)\delta_{y}}} \right\rbrack -} \\{{\left( {\theta_{z1} - \theta_{z2}} \right)^{2}\left( {\frac{1}{2} - \frac{\xi}{2} - \frac{\xi^{2}}{4}} \right)} - {\left( {\theta_{y1} - \theta_{y2}} \right)^{2}\left( {\frac{1}{2} - \frac{\xi}{2} - \frac{\xi^{2}}{4}} \right)} -} \\{{\frac{1}{4}\left( {\alpha_{z} + \frac{\beta_{z}}{2} - {\xi\;\frac{\delta_{z}}{2}}} \right)^{2}} - {\frac{1}{4}\left( {\alpha_{y} + \frac{\beta_{y}}{2} - {\xi\;\frac{\delta_{y}}{2}}} \right)^{2}} -} \\{{\frac{1}{8}\left( {\beta_{z}^{2} + \beta_{y}^{2}} \right)} + {\frac{\xi^{2}}{8}\left( {\delta_{z}^{2} + \delta_{y}^{2}} \right)} + \ldots}\end{bmatrix}.}\end{matrix} & (14)\end{matrix}$Thus, to account for contributions to the optical path difference due todeviations of the input and/or measurement beam paths from a nominalpath, Equation (2) can be re-expressed asΦ=2pkLζ[1−(θ_(z)−η_(z))²−(θ_(y)−η_(y))²]+2k(X ₁ +X ₂)  (15)where terms are retained up to quadratic order and the terms arisingfrom the reference beam path have been omitted. The terms arising fromthe reference beam may be added as required according the describedprocedure.

The systematic effects of departures of surface 190F from a planesurface 190A on the direction of the measurement axis as represented byη_(z) and η_(y) given by Equations (12) and (13) are dependent on theapproximate second order spatial derivative of the profile of surface190F, i.e., (θ_(z1)−θ_(z2)) and (θ_(y1)−θ_(y2)). This is to becontrasted with the systematic effect of a rotation of mirror object 190which depends on the first order spatial derivative or gradient of thesurface 190A, e.g., θ_(z) and θ_(y) The lack of symmetry with respect tothe two systematic effects is because changes in θ_(z) and θ_(y)represent rotations of mirror object 190 as a solid body while therotation specified for example by θ_(z1) represents a local rotation ofa portion of mirror object 190. The lack of symmetry with respect to thetwo systematic effects may also be understood as associated with therespective different points of rotation of mirror object 190, e.g., thepoint of rotation associated with θ_(z) and θ_(y) is at the intersectionof the measurement axis and surface 190A and the point of local rotationassociated with θ_(z1) for example, corresponds to the intersection ofthe path of first pass measurement beam 22 with surface 190F.

The systematic effects of the approximate second order spatialderivative of surface 190F can result in high precision specificationsfor the surface of mirror objects in certain end use applications. As anexample, consider an application where the desired accuracy of a lineardisplacement measurement is 0.1 nm, the value of the measurement pathL=0.7 m, ξ=1.1, and θ_(z)=0.5 millirad. For a deformation of surface190F with an amplitude a and a spatial wavelength Λ=1 cm, the subsequentspecification on a isa≦0.2 nm  (16)or a≦λ/3000 for λ=633 nm. The effects of spatial wavelengths greaterthan or of the order of the 1/e² diameter of the measurement beams willnot be eliminated by integration over the photosensitive area ofdetector 120. A general expression for the specification on theamplitude a in terms L, θ=(θ_(z) ²+θ_(y) ²)^(1/2), a separation dbetween the first and second pass measurement beams 22 and 24, ξ, and Λfor an error of ε in a linear displacement is

$\begin{matrix}{a \leq {\left( \frac{ɛ}{4\;\pi\;\xi\; L\;\theta} \right)\;{\left( \frac{\Lambda}{\sin\;\left( {\pi\;{d/\Lambda}} \right)} \right).}}} & (17)\end{matrix}$

Note that while errors δ_(z) and δ_(y) may effect the magnitude of theheterodyne signal of signal 52 because these errors are related to theextent to which the measurement and reference beam components in theoutput beam overlap, the magnitude of the heterodyne signal alone maynot be an accurate indication of beam path deviations. This is becausealthough δ_(z) and δ_(y) may be substantially zero, other components ofζ, η_(z), and η_(y), may still contribute to the optical pathdifference.

Deviations α_(z) and α_(y) are typically a function of the stability andalignment of source 115 and a respective beam system. Angleinterferometer 150 (shown in FIG. 1) provides a measure of α_(z), whilea similar angular displacement interferometer (not shown) orientedorthogonally to angle interferometer 150 can provide a measure of α_(y).In embodiments where source 115 and beam delivery system aresufficiently stable, interferometry system 100 need not includesubsystem 151, and α_(z) and α_(y) can be re-calibrated as necessaryduring periodic system maintenance.

Referring to FIG. 4-FIG. 8, one example of an angle interferometer isinterferometer 700 which makes angle measurements in one plane of theaverage direction of propagation of beam 712 relative to a predefinedoptical axis. Angle interferometer 700 includes beam-shearing assemblygenerally shown at element numeral 830, analyzer 840, lens 846, detector860, and electronic processor 870. For heterodyne interferometry, inputbeam 712 includes two orthogonally polarized optical beam componentshaving a difference in frequencies of f₁. The planes of polarization ofthe two orthogonally polarized components are parallel and orthogonal tothe plane of FIG. 4, respectively.

Beam-shearing assembly 830 introduces a lateral shear S_(a1) between thetwo orthogonally polarized beams 850 and 852, respectively (see FIG. 4).A portion of each of the spatially sheared output beams 850 and 852 aretransmitted by analyzer 840 as components 854 and 856, respectively.Analyzer 840 is orientated so that beam components 854 and 856 are bothpolarized in a common plane orientated at 45 degrees to the plane ofFIG. 4.

Next, beam components 854 and 856 are incident on lens 846 wherein lens846 focuses beam components 854 and 856 to spots on detector 860 to bedetected preferably by a quantum photon detector to generate electricalinterference signal 862 or heterodyne signal s₁. The spots substantiallyoverlap. Heterodyne signal s₁ is transmitted to electronic processor 870for determination of the heterodyne phase of signal s₁ and acorresponding average direction of propagation of beam 712 in the planeof FIG. 4.

Beam-shearing assembly 830 includes polarizing beam-splitters 832 and838, right angle prisms 833 and 837, and truncated Porro prisms 835 and836. The component of beam 712 polarized in the plane of FIG. 4 istransmitted by polarizing beam-splitter 832, reflected by right angleprism 833, redirected by truncated Porro prism 836, and reflected bypolarizing beam-splitter 838 as beam 850. The component of beam 712polarized orthogonal to the plane of FIG. 4 is reflected by polarizingbeam-splitter 832, redirected by truncated Porro prism 835, reflected byright angle prism 837, and transmitted by polarizing beam-splitter 838as beam 852.

Note that the optical path in glass for each of beams 854 and 856through beam-shearing assembly 830 and analyzer 840 are preferably thesame. This feature of the apparatus design of the first embodimentproduces a high stability interferometer system with respect to changesin temperature.

Heterodyne signal s₁ may be written ass ₁ =A ₁ cos(ω₁ t+φ ₁+ζ₁)  (18)whereφ₁=2k ₁ n[d ₁ cos θ′₁ +d ₂ cos θ′₂ −d ₃ cos θ′₃ −d ₄ cos θ′₄],  (19)ω₁=2 πf₁, ζ₁ is an offset phase not associated with phase φ₁, k₁=2π/λ₁,λ₁ is the wave length of input beam 712, θ′₁ and θ′₂ are angles ofincidence of beam 850 at right angle prism 833 and at the polarizingbeam-splitter 838, respectively, θ′₃ and θ′₄ are angles of incidence ofbeam 852 at polarizing beam-splitter 832 and at right angle prism 837,respectively, and d₁, d₂, d₃, and d₄ are defined in FIG. 5. It has beenassumed in Equation (19) for the purposes of demonstrating the featuresof the present invention in a simple fashion without departing from thescope and spirit of the present invention that all of the optical pathsin beam-shearing assembly 30 have the same index of refraction. For anon-limiting example of d₁=d₃, d₂=d₄, θ′₁+θ′₂=π/2, and θ′₃+θ′₄=π/2,Equation (19) reduces to the simpler expression for φ₁,

$\begin{matrix}{\varphi_{1} = {2^{1/2}k_{1}{{n\begin{bmatrix}{{\left( {d_{1} - d_{2}} \right)\left\lbrack {{\cos\left( {\theta_{1}^{\prime} + {\pi/4}} \right)} + {\cos\left( {\theta_{4}^{\prime} + {\pi/4}} \right)}} \right\rbrack} +} \\{\left( {d_{1} + d_{2}} \right)\left\lbrack {{\sin\left( {\theta_{1}^{\prime} + {\pi/4}} \right)} - {\sin\left( {\theta_{4}^{\prime} + {\pi/4}} \right)}} \right\rbrack}\end{bmatrix}}.}}} & (20)\end{matrix}$Lateral shear S_(a1) is related to properties of beam-shearing assembly830 according to the equation

$\begin{matrix}{S_{a1} = {2\begin{bmatrix}{{\left( {{d_{1}\mspace{11mu}\sin\mspace{11mu}\theta_{1}^{\prime}} - {d_{2}\mspace{11mu}\sin\mspace{11mu}\theta_{2}^{\prime}}} \right)\mspace{11mu}\sec\mspace{11mu}\phi_{1}^{\prime}\mspace{11mu}\cos\mspace{11mu}\phi_{1}} +} \\{\left( {{d_{3}\mspace{11mu}\sin\mspace{11mu}\theta_{3}^{\prime}} - {d_{4}\mspace{11mu}\sin\mspace{11mu}\theta_{4}^{\prime}}} \right)\mspace{11mu}\sec\mspace{11mu}\phi_{3}^{\prime}\mspace{11mu}\cos\mspace{11mu}\phi_{3}}\end{bmatrix}}} & (21)\end{matrix}$where φ₁ and φ′₁ are the angles of incidence and refraction of beam 850at entrance facet of polarizing beam-splitter 832 and φ₃ and φ′₃ are theangles of incidence and refraction of beam 852 at entrance facet ofpolarizing beam-splitter 832 (see FIG. 5). For the non-limiting example,

$\begin{matrix}{S_{a1} = {2^{1/2}{\begin{Bmatrix}{{\left( {d_{1} - d_{2}} \right)\begin{bmatrix}{{{\sin\left( {\theta_{1}^{\prime} + {\pi/2}} \right)}\sec\;\phi_{1}^{\prime}\cos\;\phi_{1}} +} \\{{\sin\left( {\theta_{4}^{\prime} + {\pi/2}} \right)}\sec\;\phi_{3}^{\prime}\cos\;\phi_{3}}\end{bmatrix}} +} \\{\left( {d_{1} + d_{2}} \right)\begin{bmatrix}{{{\sin\left( {\theta_{1}^{\prime} - {\pi/2}} \right)}\sec\;\phi_{1}^{\prime}\cos\;\phi_{1}} -} \\{{\sin\left( {\theta_{4}^{\prime} - {\pi/2}} \right)}\sec\;\phi_{3}^{\prime}\cos\;\phi_{3}}\end{bmatrix}}\end{Bmatrix}.}}} & (22)\end{matrix}$

The expression given for S_(a1) by Equations (21) and (22) represent theprimary mechanism used for generation of the beam shear. However, thereare other mechanisms for introducing a beam shear such as associatedwith angle of incidence dependent phase shifts (e.g., Goos-Häncheneffect).

Amplitude A₁ is proportional to a good approximation to a Fouriercomponent of the Fourier transform of |h(p₁)|², i.e.,A ₁ ∝∫|h(p ₁)|² cos [4k ₁ p ₁ S ₁ ]dp ₁  (23)where h (p₁) is the Fourier transform of the amplitude of one of thebeams 854 or 856 at lens 846 multiplied by the pupil function of lens846,p _(j)=sin θ_(o,j)+sin θ_(i,j) j,=1, 2 . . . ,  (24)and the definition of θ_(o,j) and θ_(i,j) are shown in FIG. 6. Anglesθ_(o,j) and θ_(i,j) are conjugate angles of principle rays of beam j inthe object and image space of lens 846. The definition of p_(j) is shownin FIG. 7.

It is evident from Equations (19) and (20) that the resolution of phaseφ₁ in terms of a change in a direction of an optical beam is increasedas the length 2^(3/2)(d₁−d₂) is increased. However, the usable range for2^(3/2)(d₁−d₂) is defined by the spatial frequency bandwidth of theFourier transform of |h(p₁)|² as shown by Equation (23).

The optimum value for 2^(3/2)(d₁-d₂) is generally equal to approximatelyone half a characteristic spatial dimension of a beam transmitted by arespective pupil. Consider, for example, the case of a rectangular pupilof dimension b in the plane of FIG. 4 for both beam 854 and beam 856 atlens 846 and the amplitudes of beams 854 and 856 being uniform acrossrespective pupils. For this case, |h(p₁)|² is a sinc function squared,i.e., (sin x/x)², and the Fourier transform of |h(p₁)|² is a trianglefunction, Δ. Triangle function, Δ, has a maximum value of 1 for2^(3/2)(d₁−d₂)=0 and has a value of 0 for 2^(3/2)(d₁−d₂)≧b. Therefore,amplitude A₁=0 for 2^(3/2)(d₁−d₂)≧b and the resolution of phase φ₁ interms of a change in a direction of an optical beam is 0 for2^(3/2)(d₁−d₂)=0. Thus the optimum value for 2^(3/2)(d₁−d₂) is in thiscase approximately b/2. The actual optimum value for 2^(3/2)(d₁−d₂) willdepend on the criterion used to define an optimum operating conditionwith respect to a signal-to-noise ratio, for example. For the case wherethe components of beam 712 have Gaussian intensity profiles, the optimumvalue for 2^(3/2)(d₁−d₂) will be approximately w where w is the radiusat which the intensity of beam 712 has a value equal to 1/e of theintensity at beam 712 at its center.

For an example of a beam having a Gaussian intensity profile with 2w=5.0mm, θ₁=45 degrees, and λ₁=633 nm, the sensitivity of the phase φ₁ tochanges in dφ₁ and dφ₃ expressed in differential form is given by theequation

$\begin{matrix}\begin{matrix}{{d\;\varphi_{1}} = {k_{1}{w\left\lbrack \frac{{d\;\phi_{1}} + {d\;\phi_{3}}}{2} \right\rbrack}}} \\{= {{- 2.5} \times {{10^{4}\left\lbrack \frac{{d\;\phi_{1}} + {d\;\phi_{3}}}{2} \right\rbrack}.}}}\end{matrix} & (25)\end{matrix}$

Note, as evident from Equation (25), that the sensitivity of the changein phase φ₁ with respect to changes in angles dφ₁ and dφ₃ is independentof the index of refraction n. This is an important property of the firstembodiment of the angle interferometer. In particular, the sensitivityof the change in phase φ₁ with respect to changes in angles dφ₁ and dφ₃has a sensitivity to temperature changes that is independent in firstorder to thermal induced changes in the refractive index of the opticalelements of beam-shearing assembly 830 and only dependent on thermalcoefficients of expansion of the optical elements of beam-shearingassembly 830. The thermal coefficients of the elements of beam-shearingassembly 830 can be selected to be less than ≦0.5 ppm/° C. For similarreasons, the zero value of φ₁ also exhibits a corresponding lowsensitivity to changes in temperature of beam-shearing assembly 830.

The two primary quantities that place restrictions on the range ofaverage value [dφ₁+dφ₃]/2 that can be accommodated by the firstembodiment are the magnitude of the difference [dφ₁−dφ₃]/2 and the sizeof the sensitive area of detector 860. The amplitude of the heterodynesignal will be reduced by a factor of approximately 2 when

${{wk}_{1}\left\lbrack \frac{\left\lbrack {{d\;\phi_{1}} - {d\;\phi_{3}}} \right\rbrack}{2} \right\rbrack} \approx 1.$The higher terms in dφ₁ and dφ₃ that are omitted in Equation (25) can beeasily determined from Equation (19) if required for a particular enduse application.

Another embodiment of beam-shearing assembly 830 is showndiagrammatically in FIG. 8 and includes two prisms 8330 and 8332 andpolarization beam-splitter interface 8340. A first component of inputbeam 712 is transmitted twice by polarization beam-splitter interface8340 and reflected by facets of prisms 8330 and 8332 to form output beam8350. A second component of input beam 712 is reflected twice bypolarization beam-splitter interface 8340 and reflected by facets ofprisms 8330 and 8332 to form beams 8350 and 8352.

The two prisms 8330 and 8332 and polarization beam-splitter interface8340 exhibit properties the same as a Penta prism with respect torelationship of the direction of propagation of beam 712 and thedirections of propagation for beams 8350 and 8352. Prisms 8330 and 8332are preferably isomorphic with relative sizes selected to introduce abeam shear S_(a3) between beams 8350 and 8352. The optical paths inrefractive media are substantially the same for beam 8350 and 8352. Theremaining descriptions of beams 8350 and 8352 are the same as thecorresponding portion of the descriptions given for beams 850 and 852 ofthe first embodiment with shear S_(a1) replaced by shear S_(a3).

Details of additional angular displacement interferometers are disclosedin PCT Publication WO 00/66969 by Henry A. Hill and published Nov. 9,2000, the contents of which is incorporated herein by reference, and inU.S. patent application Ser. No. 10/271,034 by Henry A. Hill, filed Oct.15, 2002 and entitled “INTERFEROMETER FOR MEASURING CHANGES IN OPTICALBEAM DIRECTIONS.”

Referring again to FIG. 2 and FIG. 3, deviations θ_(z1), θ_(z2), θ_(y1),θ_(y2), β_(z), β_(y), δ_(z), and δ_(y) depend on HSPMI 111 andmeasurement object 190. Due to the inhomogeneous nature of the defectsgiving rise to beam path deviations, deviations θ_(z1), θ_(z2), θ_(y1),θ_(y2), β_(z), β_(y), δ_(z), and δ_(y) can vary for different nominalpaths. Moreover, deviations θ_(z1), θ_(z2), θ_(y1), θ_(y2), β_(z),β_(y), δ_(z), and δ_(y) can vary as a function of the propagationdirection of the input beam. Accordingly, deviations θ_(z1)θ_(z2),θ_(y1), θ_(y2), β_(z), β_(y), δ_(z), and δ_(y) can be parameterized as afunction of measurement object displacement relative to theinterferometer, the angular orientation of the measurement object, andthe input beam propagation direction.

In applications where these deviations typically change slowly withtime, such as, for example, in many precision metrology applications,deviations can be determined prior to deployment of the interferometerin its end use application. This data can be stored in a representation,such as a lookup table, which is accessed when the interference phasedata captured using system 100 is to be analyzed. In some embodiments,the representation relating the observable parameters to beam deviationscan be in the form of a functional representation (e.g., one or morealgebraic functions), and the beam deviations can be determined from theparameters using the functions.

In certain embodiments, measurement object displacement can bedetermined using an iterative process. Where the correction data isparameterized by interferometrically determined parameters (e.g.,measurement object displacement and/or angular orientation), the systemcan iterate the parameter and deviation term determination until thesystem converges on a value for the parameter. For example, where thedeviation data is parameterized by measurement object displacement, thesystem can make an initial determination for the displacement from themeasured phase using Equation (2). Using the initial displacement value,the system then determines the deviation terms from the representation.Using this data, the system recalculates the displacement using Equation(15) to provide a once-corrected displacement value. The system iteratesthis procedure by re-determining the deviation terms based on theonce-corrected displacement value. This process can be repeated untilthe corrected displacement suitably converges.

Data relating the observable parameters to deviation angles can becharacterized in a calibration procedure prior to installing theinterferometer and other components in their end-use application such asdescribed in U.S. application Ser. No. 10/366,587 entitled “APPARATUSAND METHOD FOR QUANTIFYING AND COMPENSATING NON-CYCLIC NON-LINEARITY ININTERFEROMETRY SYSTEMS” to Henry A. Hill, the contents of which areherein incorporated in their entirety by reference. In some embodiments,deviations β_(z), β_(y), δ_(z), and δ_(y) can be measured by splittingoff a portion of the appropriate beam with a non-polarizing beamsplitter, and monitoring the beam direction while scanning themeasurement object displacement, orientation angle, and/or direction ofthe input beam. For example, in order to determine β_(z) or β_(y) abeam-splitter can be positioned in both the first pass and second passpath of the measurement beam to the measurement object. The measurementbeam direction is then tracked for each pass by monitoring the directionof the beam directed out of the measurement beam path by thebeam-splitter while the system scans the measurement objectdisplacement, orientation angle, and/or input beam direction of theinterferometer.

Non-zero values for the deviations θ_(z1), θ_(z2), θ_(y1), θ_(y2),β_(z), β_(y), δ_(z), and δ_(y) can arise from imperfections in theinterferometer and/or plane mirror measurement object. For example,non-zero values of θ_(z1), θ_(z2), θ_(y1), and/or θ_(y2) may be causedby surface imperfections of the plane mirror measurement object, and cancause a deviation of the first and/or second pass measurement beam fromthe nominal path. Additionally, imperfections in the optical componentsmaking up the interferometer can contribute to β_(z) and/or β_(y),either prior to the measurement beam's first pass to the measurementobject or between the beam's first and second pass the measurementobject.

In certain embodiments, local surface imperfections of a plane mirrormeasurement object can be measured by monitoring the beam direction of abeam reflected from the plane mirror measurement object. Notably suchtechniques can provide the local slope information of the plane mirrormeasurement object with resolution on the order of the diameter of thereflected beam. Beam directions can be monitored interferometrically ornon-interferometrically. Examples of suitable interferometers formonitoring beam directions include angle interferometers, such as theangle interferometer described above, and Hartmann-Shackinterferometers.

A Hartmann-Shack interferometer utilizes a lenslet array, which isplaced in the path of a pair of overlapping beams to be measured. Adetector is positioned at the focal plane of the lenslet array. When thebeams are coincident and their paths are parallel to the optical axes ofthe lenslets, the beams are focused to an array of spots also coincidentwith the lenslet optical axes. However, deviations of one of the beam'sdirection cause it to be focused to a different location from thosecorresponding to the undeviated beam. The detector can track thesedeviations, and the data can be used to calculate the beam propagationdirection based on the properties and location of the lenslet array. Useof Hartmann-Shack interferometers (also termed Hartmann-Shack sensors)in other applications is disclosed, for example, by Liang, J andco-workers in “Objective measurement of wave aberrations of the humaneye with the use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc.Am. (A), 11, 1949-57 (1994), by J. Liang and D. R. Williams in“Aberrations and retinal image quality of the normal human eye,” J.Opt.Soc. Am. (A), 14, 2873-83(1997), and P. M. Prieto and co-workers in“Analysis of the performance of the Hartmann-Shack sensor in the humaneye,” J. Opt. Soc. Am. (A), 17, 1388-98 (2000).

Non-interferometric methods of monitoring beam direction include, forexample, tracking the beam direction using a pixilated detector array(e.g., a CCD or CMOS camera). As the beam direction changes, it impingeson different detector elements in the array. By tracking the location ofthe beam on the array while scanning measurement object displacement,orientation angle, and input beam direction, the system can determinevariations of the beam direction as a function of the scannedparameters. When using a pixilated detector array, the optical path ofthe tracked beam to the detector array should be sufficiently long toprovide sufficient angular resolution.

While the aforementioned deviations can be monitored directly usingtechniques disclosed herein, imperfections in components of theinterferometry system (including surface imperfections of the planemirror measurement object) can also be characterized in other ways. Forexample, imperfections in the reflecting surface of the plane mirrormeasurement object (e.g., variations in the mirror's surface topography)can be accounted for by measuring the mirror's figure, which is ameasure of a mirror's surface topography. The figure of each measurementobject can be characterized, for example, using a Fizeau interferometer.The figure of the portions of the measurement objects may also bedetermined by techniques such as described in cited commonly owned U.S.patent application ser. No. 09/853,114 entitled “IN-SITU STAGE MIRRORCHARACTERIZATION,” filed May 10, 2001, U.S. patent application Ser. No.10/217,531, also entitled “IN-SITU MIRROR CHARACTERIZATION,” filed onAug. 13, 2002, International Patent Application No. PCT/US02/25652entitled “IN-SITU STAGE MIRROR CHARACTERIZATION” and U.S. patentapplication Ser. No. 10/406,749, entitled “METHOD AND APPARATUS FORSTAGE MIRROR MAPPING,” filed Apr. 3, 2003, which claims priority toProvisional Patent Application 60/371,172 filed on Apr. 9, 2002, withthe same title. These applications name Henry Allen Hill as inventor,and the entire contents of each is hereby incorporated by reference.

In embodiments where imperfections in optical components are measureddirectly, the beam path deviations can be determined from theimperfections using known relationships between the imperfections andbeam paths. For example, ray tracing tools can be used to provide beamanticipated beam paths through an interferometer based on, e.g.,empirical data related to optical surfaces and bulk imperfections insystem components.

During error calibration and/or during use of system 100, θ_(x) andθ_(y) can be monitored interferometrically or by other methods.Interferometric methods for monitoring an orientation angle of a planemirror measurement object are well established in the art. One way tointerferometrically monitor the angular orientation of a plane mirrormeasurement object is to use two displacement measuring interferometers(e.g., two HSPMIs). Where the distance between the interferometermeasurement axes is known, the interferometers can be used to providethe measurement object orientation within a first plane defined by themeasurement axes. Angular orientation of the measurement object in aplane perpendicular to the first plane can be determined by using athird displacement measuring interferometer, wherein the thirddisplacement measuring interferometer is positioned so that itsmeasurement axis and the measurement axis of one of the otherinterferometers define a plane perpendicular to the first plane. Suchmulti-axis measurements may be performed using multiple discreteinterferometers, such as multiple HSPMIs, or using a multiple axismetrology system that monitors the location of a measurement object inmultiple degrees of freedom. Examples of multiple axis metrology systemsare disclosed in U.S. Pat. No. 6,313,918, entitled “SINGLE-PASS ANDMULTI-PASS INTERFEROMETERY SYSTEMS HAVING A DYNAMIC BEAM-STEERINGASSEMBLY FOR MEASURING DISTANCE, ANGLE, AND DISPERSION,” in U.S. patentapplication Ser. No. 10/352,616, filed Jan. 28, 2003 and entitled“MULTIPLE-PASS INTERFEROMETRY,” and in U.S. patent application Ser. No.10/351,707, filed Jan. 27, 2003 and entitled “MULTIPLE DEGREE OF FREEDOMINTERFEROMETER,” all by Henry A. Hill.

Although the foregoing techniques for reducing errors in interferometer110 are described in detail with respect to an HSPMI, the techniques maybe applied to other types of interferometer. For example, the errorreducing techniques can be applied to multiple degree of freedominterferometers, such as those referenced above. An example of amultiple degree of freedom interferometer is shown in FIG. 9, whichshows an interferometry system 400 that includes interferometer 410,which measures two degrees of freedom of a measurement object 440.Interferometer 410 includes a compound optical assembly that correspondsto two HSPMIs. In addition to interferometer 410, interferometry system400 further includes a source 412, detectors 450 and 4150, and anelectronic processor 460. One HSPMI includes polarization abeam-splitter 430, a retroreflector 432, quarter wave phase retardationplates 434 and 436, and generates first and second pass measurementbeams 422 and 424, respectively, and an output beam 426. The secondHSPMI includes polarization beam-splitter 430, a retroreflector 4132,quarter wave phase retardation plates 434 and 436, and generates firstand second pass measurement beams 4122 and 4124, respectively, andoutput beam 4126.

Input beam 420 is furnished by source 412. A non-polarizingbeam-splitter 442 splits beam 420 into two beams that correspond to theinput beam for each HSPMI. A mirror 444 directs the redirected portionof beam 420 back towards interferometer 410.

An electrical interference signal 452 is generated by the detection ofoutput beam 426 in detector 450. Detector 450 comprises a polarizer tomix the reference and measurement beam components of output beam 426with respect to polarization. Electrical interference signal 452contains a heterodyne signal having a heterodyne phase Φ₁. A secondelectrical interference signal 4152 is generated by the detection ofoutput beam 4126 in detector 4150. Detector 4150 comprises a polarizerto mix the reference and measurement beam components of output beam 4126with respect to polarization. Electrical interference signal 4152contains a heterodyne signal having a heterodyne phase Φ₂.

Heterodyne phases Φ₁ and Φ₂ can each be represented by equationscorresponding to Equations (4), (5), (6), and (11). Accordingly, theoptical path difference for corresponding to each phase can bedetermined as in the foregoing description. These optical pathdifferences are directly related to the displacement of the measurementobject with respect to the interferometer at two different locations onthe measurement object surface. These locations correspond to themidpoints of where beams 422 and 424 and beams 4122 and 4124 contact thesurface, respectively.

The two displacement measurements can be used to determine the angularorientation of the measurement object when the distance between thelocations where the displacements are measured is known. Because thesystem accounts for contributions to the optical path difference causedby beam path deviations in the two displacement measurements, the systemalso accounts for these deviations in the angular orientationmeasurement.

In general, the scale factors, ζ, will not be the same and themeasurement axes of the two interferometers corresponding to measurementbeams 422 and 424 and to measurement beams 4122 and 4124 will, ingeneral, not be parallel. The different directions for the respectivemeasurement axes are represented by two different sets of η_(z), andη_(y) for the two interferometers. The different directions for therespective measurement axes can generate systematic errors in a computedchange in orientation of measurement object 440 where the computedchange is based on observed differences in the displacements ofmeasurement object 440 uncompensated for the lack of parallelism of therespective measurement axes and an assumed constant spacing of therespective measurement axes.

In general, one or more different approaches may be adopted tocompensate measurements for the lack of parallelism of respectivemeasurement axes. In some embodiments, for example, rather thancompensate the displacement measurements for the effects of lack ofmeasurement axis parallelism, one can use a parameter, d, as a spacingbetween the measurement axes, where d is not a constant but depends, forexample, on measurement path length L. The functional dependence betweend and, for example, L, can be measured by techniques such as describedin cited U.S. patent application Ser. No. 10/366,587 and/or by the useof Equations (4), (5), and (6) or (11). In certain embodiments, themeasurement axes can be assumed to be parallel (i.e., wherein d is aconstant) and the measured displacements are individually compensatedfor the effects finite values for corresponding sets of η_(z) and η_(y).

An advantage of the compensation aspects of the present invention isthat tolerances on optical components of the interferometer that affect,for example, values of β_(z) and β_(y) can be relaxed leading to lowermanufacturing costs.

The accuracy specifications on d, η_(z), η_(y), and ζ that are generatedin certain end use applications can be high. Consider an example wherethe required accuracy of a linear displacement measurement is 0.1 nm,the value of the measurement path L=0.7 m, d=40 mm, ξ=1.1, and θ_(z)=0.5milliradian. Further assume that the accuracy ε_(θ) required for θ is

$\begin{matrix}{ɛ_{\theta} \leq \frac{\left( {0.1\mspace{14mu}{nm}} \right)}{\left( {40\mspace{14mu}{mm}} \right)} \leq {2.5\mspace{14mu}{{nanoradian}.}}} & (26)\end{matrix}$The correspond accuracy ε_(d) required for d isε_(d)≦0.2 micron.  (27)The corresponding accuracy ε_(η) required for η=(η_(z) ²+η_(y) ²)^(1/2)isε_(η)≦0.13 microradian.  (28)

The accuracy specifications on d, η_(z), η_(y), and ζ may become evenhigher when the interferometer system is used in conjunction with anoff-axis alignment scope. The effects of errors in such parameters aregenerally amplified by the ratio of the displacement of the off-axisalignment scope from the measurement axes of the interferometer systemand d. A typical value for the ratio is 4. In some embodiments,interferometer 410 can be used to provide measurements of the angularorientation of the measurement object used to look-up the appropriatebeam deviation data and determine the displacement of the measurementobject with respect to the interferometer. In other words, such angularorientation measurements can be used to determine θ_(z) and θ_(y), andto determine appropriate values for ζ, η_(z), and η_(y) in Equation(15).

In the foregoing embodiments, the optical path difference between themeasurement and reference beams is directly related to the displacementof the measurement object relative to the interferometer. However, inother embodiments, the error correction techniques described herein canbe applied to interferometers in which the optical path difference isdirectly related to other degrees of freedom of the interferometrysystem. For example, in some embodiments, the optical path differencecan be directly related to the angular orientation of the measurementobject. Such embodiments include interferometers where instead of onlythe measurement beam (not the reference beam) contacting the measurementobject, both beams are directed to contact the measurement object but atdifference locations. In such a configuration, the phase is directlyrelated to an angular orientation of the measurement object in the planedefined by the two beam paths. Examples of such interferometers aredescribed aforementioned U.S. patent application Ser. No. 10/351,708,entitled “MULTI-AXIS INTERFEROMETER,” filed Jan. 27, 2003, by Henry A.Hill.

The type of analysis described previously for an HSPMI can be applied tothese other types of interferometer. In general, one can determine arelationship corresponding to Eq. (12) based on the geometricconfiguration of the interferometer, and the system can subsequentlydetermine the angular orientation (or other degree of freedom) of themeasurement object from interferometric phase measurements based on therelationship.

More generally, examples of other forms of interferometers that mayutilize the error correction techniques disclosed herein include bothsingle and multiple pass interferometers (the HSPMI is a double passinterferometer), and include passive interferometers, dynamicinterferometers, and dispersion interferometers. Alternatively, oradditionally, the error correction techniques can be applied tointerferometers that monitor more than one degree of freedom,interferometers that monitor variations in angular orientation of ameasurement object, and angular displacement interferometers thatmeasure beam propagation direction.

Examples of dynamic interferometers are described in U.S. patentapplication Ser. No. 10/226,591 filed Aug. 23, 2002 and entitled“DYNAMIC INTERFEROMETER CONTROLLING DIRECTION OF INPUT BEAM” by Henry A.Hill. Examples of passive zero shear interferometers are described inU.S. patent application Ser. No. 10/207,314, entitled “PASSIVE ZEROSHEAR INTERFEROMETERS,” filed Jul. 29, 2002, by Henry A. Hill. Examplesof angular displacement interferometers are described in: U.S. patentapplication Ser. No. 10/226,591 entitled “DYNAMIC INTERFEROMETERCONTROLLING DIRECTION OF INPUT BEAM,” filed Aug. 23, 2002; U.S.Provisional Application 60/314,345 filed Aug. 22, 2001 and entitled“PASSIVE ZERO SHEAR INTERFEROMETERS USING ANGLE SENSITIVEBEAM-SPLITTERS,” both by Henry A. Hill, and U.S. patent application Ser.No. 10/272,034 entitled “INTERFEROMETERS FOR MEASURING CHANGES INOPTICAL BEAM DIRECTION” and filed Oct. 15, 2002 by Henry A. Hill andJustin Kreuzer. Alternatively, or additionally, interferometry systemsmay include one or more differential angular displacementinterferometers, examples of which are also described in U.S. patentapplication Ser. No. 10/271,034. Examples of interferometry systems formeasuring more than one degree of freedom and for reducing beam shearare described in U.S. patent application Ser. No. 10/352,616 filed Jan.28, 2003 and entitled “MULTIPLE-PASS INTERFEROMETRY” by Henry A. Hill.Other forms of multiple pass interferometers are described in an articleentitled “Differential interferometer arrangements for distance andangle measurements: Principles, advantages and applications” by C.Zanoni, VDI Berichte Nr. 749, 93-106 (1989). Examples of two-wavelengthdispersion interferometers are described in U.S. Pat. No. 6,219,144 B1entitled “APPARATUS AND METHOD FOR MEASURING THE REFRACTIVE INDEX ANDOPTICAL PATH LENGTH EFFECTS OF AIR USING MULTIPLE-PASS INTERFEROMETRY”by Henry A. Hill, Peter de Groot, and Frank C. Demarest and U.S. Pat.No. 6,327,039 B1 by Peter de Groot, Henry A. Hill, and Frank C.Demarest.

The interferometry systems described herein provide highly accuratemeasurements. Such systems can be especially useful in lithographyapplications used in fabricating large scale integrated circuits such ascomputer chips and the like. Lithography is the key technology driverfor the semiconductor manufacturing industry. Overlay improvement is oneof the five most difficult challenges down to and below 100 nm linewidths (design rules), see, for example, the Semiconductor IndustryRoadmap, p. 82 (1997).

Overlay depends directly on the performance, i.e., accuracy andprecision, of the distance measuring interferometers used to positionthe wafer and reticle (or mask) stages. Since a lithography tool mayproduce $50-100 M/year of product, the economic value from improvedperformance distance measuring interferometers is substantial. Each 1%increase in yield of the lithography tool results in approximately $1M/year economic benefit to the integrated circuit manufacturer andsubstantial competitive advantage to the lithography tool vendor.

The function of a lithography tool is to direct spatially patternedradiation onto a photoresist-coated wafer. The process involvesdetermining which location of the wafer is to receive the radiation(alignment) and applying the radiation to the photoresist at thatlocation (exposure).

To properly position the wafer, the wafer includes alignment marks onthe wafer that can be measured by dedicated sensors. The measuredpositions of the alignment marks define the location of the wafer withinthe tool. This information, along with a specification of the desiredpatterning of the wafer surface, guides the alignment of the waferrelative to the spatially patterned radiation. Based on suchinformation, a translatable stage supporting the photoresist-coatedwafer moves the wafer such that the radiation will expose the correctlocation of the wafer.

During exposure, a radiation source illuminates a patterned reticle,which scatters the radiation to produce the spatially patternedradiation. The reticle is also referred to as a mask, and these termsare used interchangeably below. In the case of reduction lithography, areduction lens collects the scattered radiation and forms a reducedimage of the reticle pattern. Alternatively, in the case of proximityprinting, the scattered radiation propagates a small distance (typicallyon the order of microns) before contacting the wafer to produce a 1:1image of the reticle pattern. The radiation initiates photo-chemicalprocesses in the resist that convert the radiation pattern into a latentimage within the resist.

Interferometry systems are important components of the positioningmechanisms that control the position of the wafer and reticle, andregister the reticle image on the wafer. If such interferometry systemsinclude the features described above, the accuracy of distances measuredby the systems increases as cyclic error contributions to the distancemeasurement are minimized.

In general, the lithography system, also referred to as an exposuresystem, typically includes an illumination system and a waferpositioning system. The illumination system includes a radiation sourcefor providing radiation such as ultraviolet, visible, x-ray, electron,or ion radiation, and a reticle or mask for imparting the pattern to theradiation, thereby generating the spatially patterned radiation. Inaddition, for the case of reduction lithography, the illumination systemcan include a lens assembly for imaging the spatially patternedradiation onto the wafer. The imaged radiation exposes resist coatedonto the wafer. The illumination system also includes a mask stage forsupporting the mask and a positioning system for adjusting the positionof the mask stage relative to the radiation directed through the mask.The wafer positioning system includes a wafer stage for supporting thewafer and a positioning system for adjusting the position of the waferstage relative to the imaged radiation. Fabrication of integratedcircuits can include multiple exposing steps. For a general reference onlithography, see, for example, J. R. Sheats and B. W. Smith, inMicrolithograpyh: Science and Technology (Marcel Dekker, Inc., New York,1998), the contents of which is incorporated herein by reference.

Interferometry systems described above can be used to precisely measurethe positions of each of the wafer stage and mask stage relative toother components of the exposure system, such as the lens assembly,radiation source, or support structure. In such cases, theinterferometry system can be attached to a stationary structure and themeasurement object attached to a movable element such as one of the maskand wafer stages. Alternatively, the situation can be reversed, with theinterferometry system attached to a movable object and the measurementobject attached to a stationary object.

More generally, such interferometry systems can be used to measure theposition of any one component of the exposure system relative to anyother component of the exposure system, in which the interferometrysystem is attached to, or supported by, one of the components and themeasurement object is attached, or is supported by the other of thecomponents.

An example of a lithography scanner 1100 using an interferometry system1126 is shown in FIG. 10. The interferometry system is used to preciselymeasure the position of a wafer (not shown) within an exposure system.Here, stage 1122 is used to position and support the wafer relative toan exposure station. Scanner 1100 includes a frame 1102, which carriesother support structures and various components carried on thosestructures. An exposure base 1104 has mounted on top of it a lenshousing 1106 atop of which is mounted a reticle or mask stage 1116,which is used to support a reticle or mask. A positioning system forpositioning the mask relative to the exposure station is indicatedschematically by element 1117. Positioning system 1117 can include,e.g., piezoelectric transducer elements and corresponding controlelectronics. Although, it is not included in this described embodiment,one or more of the interferometry systems described above can also beused to precisely measure the position of the mask stage as well asother moveable elements whose position must be accurately monitored inprocesses for fabricating lithographic structures (see supra Sheats andSmith Microlithography: Science and Technology).

Suspended below exposure base 1104 is a support base 1113 that carrieswafer stage 1122. Stage 1122 includes a plane mirror 1128 for reflectinga measurement beam 1154 directed to the stage by interferometry system1126. A positioning system for positioning stage 1122 relative tointerferometry system 1126 is indicated schematically by element 1119.Positioning system 1119 can include, e.g., piezoelectric transducerelements and corresponding control electronics. The measurement beamreflects back to the interferometry system, which is mounted on exposurebase 1104. The interferometry system can be any of the embodimentsdescribed previously.

During operation, a radiation beam 1110, e.g., an ultraviolet (UV) beamfrom a UV laser (not shown), passes through a beam shaping opticsassembly 1112 and travels downward after reflecting from mirror 1114.Thereafter, the radiation beam passes through a mask (not shown) carriedby mask stage 1116. The mask (not shown) is imaged onto a wafer (notshown) on wafer stage 1122 via a lens assembly 1108 carried in a lenshousing 1106. Base 1104 and the various components supported by it areisolated from environmental vibrations by a damping system depicted byspring 1120.

In other embodiments of the lithographic scanner, one or more of theinterferometry systems described previously can be used to measuredistance along multiple axes and angles associated for example with, butnot limited to, the wafer and reticle (or mask) stages. Also, ratherthan a UV laser beam, other beams can be used to expose the waferincluding, e.g., x-ray beams, electron beams, ion beams, and visibleoptical beams.

In some embodiments, the lithographic scanner can include what is knownin the art as a column reference. In such embodiments, theinterferometry system 1126 directs the reference beam (not shown) alongan external reference path that contacts a reference mirror (not shown)mounted on some structure that directs the radiation beam, e.g., lenshousing 1106. The reference mirror reflects the reference beam back tothe interferometry system. The interference signal produce byinterferometry system 1126 when combining measurement beam 1154reflected from stage 1122 and the reference beam reflected from areference mirror mounted on the lens housing 1106 indicates changes inthe position of the stage relative to the radiation beam. Furthermore,in other embodiments the interferometry system 1126 can be positioned tomeasure changes in the position of reticle (or mask) stage 1116 or othermovable components of the scanner system. Finally, the interferometrysystems can be used in a similar fashion with lithography systemsinvolving steppers, in addition to, or rather than, scanners.

As is well known in the art, lithography is a critical part ofmanufacturing methods for making semiconducting devices. For example,U.S. Pat. No. 5,483,343 outlines steps for such manufacturing methods.These steps are described below with reference to FIGS. 11( a) and11(b). FIG. 11( a) is a flow chart of the sequence of manufacturing asemiconductor device such as a semiconductor chip (e.g., IC or LSI), aliquid crystal panel or a CCD. Step 1151 is a design process fordesigning the circuit of a semiconductor device. Step 1152 is a processfor manufacturing a mask on the basis of the circuit pattern design.Step 1153 is a process for manufacturing a wafer by using a materialsuch as silicon.

Step 1154 is a wafer process that is called a pre-process wherein, byusing the so prepared mask and wafer, circuits are formed on the waferthrough lithography. To form circuits on the wafer that correspond withsufficient spatial resolution those patterns on the mask,interferometric positioning of the lithography tool relative the waferis necessary. The interferometry methods and systems described hereincan be especially useful to improve the effectiveness of the lithographyused in the wafer process.

Step 1155 is an assembling step, which is called a post-process whereinthe wafer processed by step 1154 is formed into semiconductor chips.This step includes assembling (dicing and bonding) and packaging (chipsealing). Step 1156 is an inspection step wherein operability check,durability check and so on of the semiconductor devices produced by step1155 are carried out. With these processes, semiconductor devices arefinished and they are shipped (step 1157).

FIG. 11( b) is a flow chart showing details of the wafer process. Step1161 is an oxidation process for oxidizing the surface of a wafer. Step1162 is a CVD process for forming an insulating film on the wafersurface. Step 1163 is an electrode forming process for formingelectrodes on the wafer by vapor deposition. Step 1164 is an ionimplanting process for implanting ions to the wafer. Step 1165 is aresist process for applying a resist (photosensitive material) to thewafer. Step 1166 is an exposure process for printing, by exposure (i.e.,lithography), the circuit pattern of the mask on the wafer through theexposure apparatus described above. Once again, as described above, theuse of the interferometry systems and methods described herein improvethe accuracy and resolution of such lithography steps.

Step 1167 is a developing process for developing the exposed wafer. Step1168 is an etching process for removing portions other than thedeveloped resist image. Step 1169 is a resist separation process forseparating the resist material remaining on the wafer after beingsubjected to the etching process. By repeating these processes, circuitpatterns are formed and superimposed on the wafer.

The interferometry systems described above can also be used in otherapplications in which the relative position of an object needs to bemeasured precisely. For example, in applications in which a write beamsuch as a laser, x-ray, ion, or electron beam, marks a pattern onto asubstrate as either the substrate or beam moves, the interferometrysystems can be used to measure the relative movement between thesubstrate and write beam.

As an example, a schematic of a beam writing system 1200 is shown inFIG. 12. A source 1210 generates a write beam 1212, and a beam focusingassembly 1214 directs the radiation beam to a substrate 1216 supportedby a movable stage 1218. To determine the relative position of thestage, an interferometry system 1220 directs a reference beam 1222 to amirror 1224 mounted on beam focusing assembly 1214 and a measurementbeam 1226 to a mirror 1228 mounted on stage 1218. Since the referencebeam contacts a mirror mounted on the beam focusing assembly, the beamwriting system is an example of a system that uses a column reference.Interferometry system 1220 can be any of the interferometry systemsdescribed previously. Changes in the position measured by theinterferometry system correspond to changes in the relative position ofwrite beam 1212 on substrate 1216. Interferometry system 1220 sends ameasurement signal 1232 to controller 1230 that is indicative of therelative position of write beam 1212 on substrate 1216. Controller 1230sends an output signal 1234 to a base 1236 that supports and positionsstage 1218. In addition, controller 1230 sends a signal 1238 to source1210 to vary the intensity of, or block, write beam 1212 so that thewrite beam contacts the substrate with an intensity sufficient to causephotophysical or photochemical change only at selected positions of thesubstrate.

Furthermore, in some embodiments, controller 1230 can cause beamfocusing assembly 1214 to scan the write beam over a region of thesubstrate, e.g., using signal 1244. As a result, controller 1230 directsthe other components of the system to pattern the substrate. Thepatterning is typically based on an electronic design pattern stored inthe controller. In some applications the write beam patterns a resistcoated on the substrate and in other applications the write beamdirectly patterns, e.g., etches, the substrate.

An important application of such a system is the fabrication of masksand reticles used in the lithography methods described previously. Forexample, to fabricate a lithography mask an electron beam can be used topattern a chromium-coated glass substrate. In such cases where the writebeam is an electron beam, the beam writing system encloses the electronbeam path in a vacuum. Also, in cases where the write beam is, e.g., anelectron or ion beam, the beam focusing assembly includes electric fieldgenerators such as quadrapole lenses for focusing and directing thecharged particles onto the substrate under vacuum. In other cases wherethe write beam is a radiation beam, e.g., x-ray, UV, or visibleradiation, the beam focusing assembly includes corresponding optics andfor focusing and directing the radiation to the substrate.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention. Otherembodiments are within the scope of the claims.

1. A method, comprising: using an interferometer in an interferometrysystem to produce an output beam comprising a phase related to anoptical path difference between a path of a first beam and a path of asecond beam, wherein the first beam contacts a measurement object at afirst location and the first or second beam contacts the measurementobject at a second location, and wherein the first and second locationsare different; providing precalibrated information that accounts forcontributions to the optical path difference caused by a deviation ofthe path of the first or second beam from a nominal beam path due to animperfection of the measurement object at the first location and due toan imperfection of the measurement object at the second location; anddetermining a position of the measurement object with respect to atleast one degree of freedom based on information derived from the outputbeam and the precalibrated information.
 2. The method of claim 1,wherein the precalibrated information accounts for contributions to theoptical path difference caused by a deviation of the path of the firstor second beam within a plane defined by the nominal beam path due tothe imperfection of the measurement object at the first or secondlocation.
 3. The method of claim 2, wherein the precalibratedinformation accounts for contributions to the optical path differencecaused by a deviation of the path of the first or second beam out of aplane defined by a nominal beam path due to the imperfection of themeasurement object at the first or second location.
 4. The method ofclaim 1, wherein the precalibrated information further accounts forcontributions to the optical path difference caused by a deviation ofthe path of the first or second beam from the nominal beam path due toan imperfection in at least one optic of the interferometer differentfrom the measurement object.
 5. The method of claim 4, wherein theimperfection in at least one optic of the interferometer comprises animperfection in a surface of the optic.
 6. The method of claim 4,wherein the imperfection in at least one optic of the interferometercomprises a bulk imperfection in the optic.
 7. The method of claim 1,wherein the precalibrated information further accounts for contributionsto the optical path difference caused by a deviation of the path of thefirst or second beam from the nominal beam path due to an imperfectionin a light source that causes an input beam derived from the lightsource to deviate from an input beam path to the interferometer.
 8. Themethod of claim 1, wherein the first or second beam contacts themeasurement object at one or more additional locations different fromthe first and second locations and the precalibrated informationaccounts for contributions to the optical path difference caused by adeviation of the path of the first or second beam from the nominal beampath due to an imperfection of the measurement object at one or moreadditional locations.
 9. The method of claim 1, wherein theprecalibrated information is parameterized in terms of at least one ofan angular orientation of the measurement object relative to theinterferometer, a distance between the measurement object and theinterferometer, and a direction of an input beam to the interferometer.10. The method of claim 9, wherein the precalibrated information isparameterized in terms of at least two of an angular orientation of themeasurement object relative to the interferometer, a distance betweenthe measurement object and the interferometer, and a direction of aninput beam to the interferometer.
 11. The method of claim 10, whereinthe precalibrated information is parameterized in terms of an angularorientation of the measurement object relative to the interferometer, adistance between the measurement object and the interferometer, and adirection of an input beam to the interferometer.
 12. The method ofclaim 9, wherein the precalibrated information is stored as arepresentation in an electronic storage medium.
 13. The method of claim12, wherein the representation comprises a lookup table.
 14. The methodof claim 12, wherein the representation comprises a functionalrepresentation.
 15. The method of claim 1, wherein the determinedposition of the measurement object with respect to at least one degreeof freedom is related to a displacement of the measurement objectrelative to the interferometer.
 16. The method of claim 15, wherein thedetermined position of the measurement object with respect to at leastone degree of freedom is the displacement of the measurement objectrelative to the interferometer.
 17. The method of claim 1, wherein thedetermined position of the measurement object with respect to at leastone degree of freedom is related to an angular orientation of themeasurement object.
 18. The method of claim 17, wherein the determinedposition of the measurement object with respect to at least one degreeof freedom is the angular orientation of the measurement object.
 19. Themethod of claim 1, wherein determining the position of the measurementobject comprises measuring the phase of the output beam and relating thephase to the position of the measurement object based on one or morevalues derived from the predetermined information.
 20. The method ofclaim 19, wherein the values derived from the predetermined informationare selected based on the phase.
 21. The method of claim 19, wherein thevalues derived from the predetermined information are selected based onan angular orientation of the measurement object.
 22. The method ofclaim 19, wherein the values derived from the predetermined informationare selected based on a path of an input beam derived from the lightsource to the interferometer relative to the nominal path.
 23. Themethod of claim 22, further comprising monitoring deviations of theinput beam path from the nominal path.
 24. The method of claim 19,wherein a relationship between the phase, Φ, and the optical pathdifference can be expressed by the equationΦ=2pkLζ(1−(θ₁−η₁)²−(θ₂−η₂)²+2k(X ₁ +X ₂), wherein p is an integer, k isa wavenumber, L is a relative distance between the interferometer andthe measurement object, θ₁ and θ₂ are angular orientation of themeasurement object with respect to the interferometer along orthogonalcoordinates, and ζ and η₁ and η₂ are terms that depend on the deviationof at least one of the beam paths from the nominal beam path, and X₁ andX₂ is are local displacements of a surface of the measurement objectfrom a nominal plane surface at the first and second locations,respectively.
 25. The method of claim 1, wherein using theinterferometer to produce the output beam comprises producing the outputbeam as the measurement object is moved relative to the interferometer,and wherein determining the position of the measurement object comprisesmonitoring the position of the measurement object during the relativemovement.
 26. The method of claim 1, wherein using the interferometer toproduce the output beam comprises separating an input beam into at leastthe first and second beams, directing the first and second beams alongtheir respective paths, and recombining the two beams after one or bothof the beams contacts the measurement object.
 27. The method of claim26, wherein the second beam contacts the measurement object at thesecond location and the optical path difference is related to an angularorientation of the measurement object with respect to theinterferometer.
 28. A lithography method for use in fabricatingintegrated circuits on a wafer, the method comprising: supporting thewafer on a moveable stage; imaging spatially patterned radiation ontothe wafer; adjusting the position of the stage; and monitoring theposition of the stage using the method of claim
 1. 29. A method forfabricating integrated circuits, the method comprising the lithographymethod of claim
 28. 30. A lithography method for use in the fabricationof integrated circuits comprising: directing input radiation through amask to produce spatially patterned radiation; positioning the maskrelative to the input radiation; monitoring the position of the maskrelative to the input radiation using the method of claim 1; and imagingthe spatially patterned radiation onto a wafer.
 31. A method forfabricating integrated circuits, the method comprising the lithographymethod of claim
 30. 32. A lithography method for fabricating integratedcircuits on a wafer comprising: positioning a first component of alithography system relative to a second component of a lithographysystem to expose the wafer to spatially patterned radiation; andmonitoring the position of the first component relative to the secondcomponent using the method of claim
 1. 33. A method for fabricatingintegrated circuits, the method comprising the lithography method ofclaim
 32. 34. A method for fabricating a lithography mask, the methodcomprising: directing a write beam to a substrate to pattern thesubstrate; positioning the substrate relative to the write beam; andmonitoring the position of the substrate relative to the write beamusing the interferometry method of claim
 1. 35. A method, comprising:using an interferometer to produce an output beam comprising a phaserelated to an optical path difference between a path of a first beam anda path of a second beam, wherein the first beam contacts a measurementobject at a first location and the first or second beam contacts themeasurement object at a second location different from the firstlocation; providing precalibrated information that accounts forcontributions to the optical path difference caused by a deviation ofthe path of the first beam out of a plane defined by a nominal beam pathdue to an imperfection of the measurement object at the first location;and determining a position of the measurement object with respect to atleast one degree of freedom based on information derived from the outputbeam and the precalibrated information.
 36. The method of claim 35,wherein the precalibrated information accounts for contributions to theoptical path difference caused by a deviation of a path of the firstbeam out within the plane defined by the nominal beam path due to theimperfection of the measurement object at the first location.
 37. Themethod of claim 35, wherein the precalibrated information accounts forcontributions to the optical path difference caused by a localdisplacement of a surface of the measurement object from a nominal planesurface at the first location.
 38. The method of claim 28, wherein theprecalibrated information accounts for contributions to the optical pathdifference caused by a deviation of the path of the first or second beamfrom a nominal beam path due to an imperfection of the measurementobject at the second location.
 39. A method, comprising: using aninterferometer to produce an output beam comprising a phase related toan optical path difference between a first beam path and a second beampath, wherein the first or second beam contacts a measurement object;providing precalibrated information that accounts for contributions tothe optical path difference caused by a deviation of a path of the firstbeam from a nominal beam path due to an imperfection of the measurementobject, and accounts for contributions to the optical path differencecaused by a deviation of the path of the first beam from the nominalbeam path due to an imperfection in one or more optics of theinterferometer different from the measurement object or in a lightsource used to produce the output beam; and determining a position ofthe measurement object with respect to at least one degree of freedombased on information derived from the output beam and the precalibratedinformation.
 40. A method, comprising: using a first interferometer anda second interferometer in an interferometry system to produce a firstoutput beam and a second output beam, respectively, wherein each outputbeam comprises a phase related to an optical path difference between twobeam paths, at least one of which contacts a measurement object;providing precalibrated information that accounts for a misalignment ofan axis of the first interferometer relative to an axis of the secondinterferometer; and determining a position of the measurement objectwith respect to at least one degree of freedom based on informationderived from the first and second output beams and the precalibratedinformation.
 41. The method of claim 40, wherein the degree of freedomcorresponds to an angular orientation of the measurement object.
 42. Themethod of claim 40, wherein for each interferometer, the precalibratedinformation accounts for contributions to the optical path differencecaused by a deviation of at least one of the beam paths from a nominalbeam path due to other imperfections in the interferometry system. 43.The method of claim 42, wherein other imperfections comprise animperfection in at least one optic of the interferometer, animperfection in the measurement object, or an imperfection in a lightsource that causes an input beam derived from the light source todeviate from an input beam path for the interferometer.
 44. An apparatuscomprising: an interferometer configured to produce an output beamcomprising a phase related to an optical path difference between a pathof a first beam and a path of a second beam, wherein the first beamcontacts a measurement object at a first location and the first orsecond beam contacts the interferometer at a second location, andwherein the first and second locations are different; and an electroniccontroller coupled to the interferometer, wherein during operation theelectronic controller determines a position of the measurement objectwith respect to at least one degree of freedom based on informationderived from the output beam and precalibrated information that accountsfor contributions to the optical path difference caused by a deviationof at least one of the beam paths from a nominal beam path due to animperfection of the measurement object at the first location and due toan imperfection of the measurement object at the second location.
 45. Alithography system for use in fabricating integrated circuits on awafer, the system comprising: a stage for supporting the wafer; anillumination system for imaging spatially patterned radiation onto thewafer; a positioning system for adjusting the position of the stagerelative to the imaged radiation; and the apparatus of claim 44 formonitoring the position of the wafer relative to the imaged radiation.46. A method for fabricating integrated circuits, the method comprising:applying a resist to a wafer; forming a pattern of a mask in the resistby exposing the wafer to radiation using the lithography system of claim45; and producing an integrated circuit from the wafer.
 47. Alithography system for use in fabricating integrated circuits on awafer, the system comprising: a stage for supporting the wafer; and anillumination system including a radiation source, a mask, a positioningsystem, a lens assembly, and the apparatus of claim 44, wherein duringoperation the source directs radiation through the mask to producespatially patterned radiation, the positioning system adjusts theposition of the mask relative to the radiation from the source, the lensassembly images the spatially patterned radiation onto the wafer, andthe apparatus monitors the position of the mask relative to theradiation from the source.
 48. A method for fabricating integratedcircuits, the method comprising: applying a resist to a wafer; forming apattern of a mask in the resist by exposing the wafer to radiation usingthe lithography system of claim 47; and producing an integrated circuitfrom the wafer.
 49. A beam writing system for use in fabricating alithography mask, the system comprising: a source providing a write beamto pattern a substrate; a stage supporting the substrate; a beamdirecting assembly for delivering the write beam to the substrate; apositioning system for positioning the stage and beam directing assemblyrelative one another; and the apparatus of claim 44 for monitoring theposition of the stage relative to the beam directing assembly.
 50. Anapparatus, comprising: an interferometer configured to produce an outputbeam comprising a phase related to an optical path difference between apath of a first beam and a path of a second beam, wherein the first beamcontacts a measurement object at a first location and the first orsecond beam contacts the measurement object at a second location,wherein the first and second locations are different; and an electroniccontroller coupled to the interferometer, wherein during operation theelectronic controller determines a position of the measurement objectwith respect to at least one degree of freedom based on informationderived from the output beam and precalibrated information that accountsfor contributions to the optical path difference caused by a deviationof the path of the first beam out of a plane defined by a nominal beampath due to an imperfection of the measurement object at the firstlocation.
 51. An interferometry system, comprising: a measurementobject; a first interferometer and a second interferometer, the firstand second interferometers respectively being configured to produce anoutput beam comprising a phase related to an optical path differencebetween a path of a first beam and a path of a second beam, wherein thefirst beam contacts the measurement object at a first location; and anelectronic controller coupled to the first and second interferometers,wherein during operation the electronic controller determines a positionof the measurement object with respect to at least one degree of freedombased on information derived from the output beam and precalibratedinformation that accounts for a misalignment of an axis of the firstinterferometer relative to an axis of the second interferometer.